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Class 11 Maths Chapter 12 Limits and Derivatives Revision Notes

The concepts of Limits and Derivatives mark the beginning of Calculus, one of the most important branches of Mathematics. Whether it is calculating the speed of a moving object, analyzing the growth of a population, optimizing engineering designs, or studying rates of change in Physics and Economics, derivatives play a crucial role. This chapter introduces the fundamental ideas of limits, the behavior of functions near a point, and the concept of the derivative as the instantaneous rate of change and slope of a tangent. These concepts form the foundation for advanced Calculus and are extensively tested in JEE Main, JEE Advanced, and CBSE Board examinations.

At ALLEN, our expert faculty have prepared these Class 11 Maths Chapter 12: Limits and Derivatives Revision Notes to help students build strong conceptual clarity in Calculus. These notes include concise theory, standard limit results, derivative formulas, shortcut techniques, solved illustrations, and previous years' question analysis. Whether you're preparing for board examinations or competitive exams, these revision notes will help you master the fundamentals of Calculus and solve problems with confidence.

1.0Chapter Snapshot

Topic	Overview
Limits	Concept and evaluation of limits
Standard Limits	Frequently used limit formulas
Left-Hand and Right-Hand Limits	Existence of limits
Continuity (Basic Idea)	Behavior of functions near a point
Derivative	Instantaneous rate of change
Geometrical Interpretation	Slope of a tangent
Algebra of Derivatives	Basic derivative rules
Applications	Foundation of Calculus and JEE problems

2.0Chapter 12 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution

Limits and Derivatives

Limits and Derivatives are fundamental concepts in calculus that lay the groundwork for more advanced topics. Understanding them is crucial for solving problems related to rates of change, optimization, and motion, among others.

1.0What are Limits and Derivatives in Mathematics?

Limits help us understand the behavior of functions as they approach a certain point. Formally, the limit of a function f(x) as x approaches a value a is the value that f(x) gets closer to as x gets closer to a. It’s expressed as:

$lim_{x \to a} f (x) = L$

Where L is the limit value.

Following fig. show the graphs of three functions. Note that in part(c), f(a) is not defined and part (b), f(a) ≠ L. But in each case, it happens at a. It is true that $lim_{x \to a} f (x) = L$ _.

Derivatives, on the other hand, measure how a function changes as its input changes. It's the rate at which the function's value changes with respect to change in the variable. The derivative of a function f(x) at a point x = a is given by:

$f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$

This formula is essentially the limit of the average rate of change as the interval shrinks to zero.

2.0Left hand Limits and Right Hand limit of a Function

$lim_{x \to a^{-}} f (x) = L$ $lim_{x \to a^{+}} f (x) = L$

$lim_{x \to a} f (x) = L$

If $lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = L (finite)$

Geometrical Meaning:

Tangent to the curve y = f(x) at a points (a, f(a)) is a limiting case of slope of secant through A.

$lim_{B \to A} m_{A B} = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$

$lim_{C \to A} m_{A c} = lim_{h \to 0} \frac{f ( a - h ) - f ( a )}{- h}$

Derivative exists if

$lim_{h \to 0} \frac{f ( a - h ) - f ( a )}{- h} = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$

3.0Limits and Derivatives Formulas

Algebra of Limits:

$lim_{x \to a} c = c$ , where c is a constant.
$lim_{x \to a} x = a$
$lim_{x \to a} [f (x) \pm g (x)] = lim_{x \to a} f (x) \pm lim_{x \to a} g (x)$
$lim_{x \to a} [f (x) \cdot g (x)] = lim_{x \to a} f (x) \cdot lim_{x \to a} g (x)$
$lim_{x \to a} \frac{f ( x )}{g ( x )} = \frac{l i m _{x \to a} f ( x )}{l i m _{x \to a} g ( x )}$
$lim_{x \to a} [(f) (x)] = lim_{x \to a} f (x)$

Limits of Some Basic Trigonometric Functions

$lim_{x \to 0} \frac{s i n x}{x} = 1$
$lim_{x \to 0} \frac{t a n x}{x} = 1$
$lim_{x \to 0} \frac{s i n ^{- 1} x}{x} = 1$
$lim_{x \to 0} \frac{t a n ^{- 1} x}{x} = 1$ (Where x is measured in radians)
If $lim_{x \to 0} f (x) = 0 t han lim_{x \to 0} \frac{s i n f ( x )}{f ( x )} = 1$ e.g. $lim_{x \to 1} \frac{s i n ( l n x )}{( l n x )} = 1$
$lim_{x \to 0} \frac{1 - c o s x}{x} = 0$

Limits of Some Exponential and Logarithmic Functions

$lim_{x \to 0} e^{x} = 1$
$lim_{x \to 0} \frac{e ^{x} - 1}{x} = 1$
$lim_{x \to \infty} (1 + \frac{1}{x})^{x} = e$
$lim_{x \to \infty} (1 + \frac{a}{x})^{x} = e^{a}$
$lim_{x \to 0} (1 + x)^{1/ x} = e$
$lim_{x \to 0} \frac{a ^{x} - 1}{x} = lo g_{e} a$
$lim_{x \to 0} \frac{l o g ( 1 + x )}{x} = 1$

xⁿFormula

$lim_{x \to a} \frac{x ^{n} - a ^{n}}{x - a} = n (a)^{n - 1}$

Basic Derivative Formulas:

$\frac{d}{d x} [c] = 0$ , where c is a constant.
$\frac{d}{d x} [x^{n}] = n x^{n - 1}$
$\frac{d}{d x} [sin x] = cos x$
$\frac{d}{d x} [cos x] = - sin x$

Algebra of Derivative of Functions:

$\frac{d}{d x} [f (x) \pm g (x)] = \frac{d}{d x} f (x) \pm \frac{d}{d x} g (x)$

$\frac{d}{d x} [f (x) \times g (x)] = f (x) \cdot \frac{d}{d x} g (x) + g (x) \cdot \frac{d}{d x} f (x)$

$\frac{d}{d x} [\frac{f ( x )}{g ( x )}] = \frac{g ( x ) \cdot \frac{d}{d x} f ( x ) - f ( x ) \cdot \frac{d}{d x} g ( x )}{( g ( x ) ) ^{2}}$

1.0Relationship Between Derivatives and Limits

The concept of derivatives is fundamentally built on the concept of limits. A derivative is essentially a limit that describes the rate of change of a function at a specific point. Without limits, the definition of a derivative wouldn't exist. When we calculate a derivative using the definition, we're actually finding the limit of the function's average rate of change as the interval over which the change is measured approaches zero.

2.0Solved Examples of Limits and Derivatives

Example 1: Evaluate the limit: $lim_{x \to 3} \frac{x ^{2} - 9}{x - 3}$

Solution:

Given: $lim_{x \to 3} \frac{x ^{2} - 9}{x - 3}$

$= lim_{x \to 3} \frac{x ^{2} - 9}{x - 3}$

Simplify the expression

$= lim_{x \to 3} \frac{( x - 3 ) ( x + 3 )}{x - 3}$

Cancelling the common factor (x – 3)

$= lim_{x \to 3} x + 3$

Now, substitute x = 3 into the simplified expression

= 3 + 3 = 6

So, $lim_{x \to 3} \frac{x ^{2} - 9}{x - 3} = 6$

Example 2: Find the derivative of f(x) = x² at x = 2.

Solution:

Using the definition of a derivative:

$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$

Substituting f(x) = x²:

$f^{'} (2) = lim_{h \to 0} \frac{( 2 + h ) ^{2} - 4}{h}$

$f^{'} (2) = lim_{h \to 0} \frac{4 + 4 h + h ^{2} - 4}{h}$

$f^{'} (2) = lim_{h \to 0} (4 + h)$

$f^{'} (2) = 4$

So, f'(2) = 4.

Example 3: Evaluate the limit: $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$

Solution:

Given: $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$

$= lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$

Simplify the expression

$= lim_{x \to 2} \frac{( x - 2 ) ( x + 2 )}{x - 2}$

Cancelling the common factor (x – 2)

$= lim_{x \to 2} x + 2$

Now, substitute x = 2 into the simplified expression

= 2 + 2 = 4

So, $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2} = 4$

Example 4: Find the derivative of f(x) = 3x³ – 5x² + 2x – 7.

Solution:

Using the power rule $\frac{d}{d x} [x^{n}] = n x^{n - 1}$ , differentiate each term:

$f^{'} (x) = 3 \cdot 3 x^{2} - 5 \cdot 2 x^{1} + 2 \cdot 1 x^{0} - 0$

$f^{'} (x) = 9 x^{2} - 10 x + 2$

Example 5: Find the derivative of f(x) = x² + 3x using the definition of a derivative.

Solution:

The derivative is given by:

$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$

Substitute f(x) = x² + 3x:

Formula to find the derivative of a function

Expand and simplify:

$f^{'} (x) = lim_{h \to 0} \frac{( x + h ) ^{2} + 3 ( x + h ) - ( x ^{2} + 3 x )}{h}$

$f^{'} (x) = lim_{h \to 0} \frac{2 x h + h ^{2} + 3 h}{h}$

Factor out h:

$f^{'} (x) = lim_{h \to 0} (2 x + h + 3)$

$f^{'} (x) = 2 x + 3$

So, the derivative is f'(x) = 2x + 3.

Example 6: Evaluate the limit $lim_{x \to \infty} \frac{5 x ^{3} - 2 x ^{2} + 4}{3 x ^{3} + x - 7}$

Solution:

Given: $lim_{x \to \infty} \frac{5 x ^{3} - 2 x ^{2} + 4}{3 x ^{3} + x - 7}$

Divide each term by x³ (the highest power of x in the denominator)

$lim_{x \to \infty} \frac{5 - \frac{2}{x} + \frac{4}{x ^{3}}}{3 + \frac{1}{x ^{2}} - \frac{7}{x ^{3}}}$

Putting the limit x = infinite

$lim_{x \to \infty} \frac{5 - \frac{2}{x} + \frac{4}{x ^{3}}}{3 + \frac{1}{x ^{2}} - \frac{7}{x ^{3}}} = \frac{5}{3}$

So, $lim_{x \to \infty} \frac{5 x ^{3} - 2 x ^{2} + 4}{3 x ^{3} + x - 7} = \frac{5}{3}$

Example 7: Evaluate $lim_{x \to 0} {\frac{1 + x - 1 - x}{x}}$

Solution:

We have

$lim_{x \to 0} {\frac{1 + x - 1 - x}{x}}$

$= lim_{x \to 0} {\frac{1 + x - 1 - x}{x} \cdot \frac{( 1 + x + 1 - x )}{( 1 + x + 1 - x )}}$

$= lim_{x \to 0} \frac{{( 1 + x ) - ( 1 - x )}}{x ( 1 + x + 1 - x )}$

$= lim_{x \to 0} \frac{2 x}{x ( 1 + x + 1 - x )}$

$= lim_{x \to 0} \frac{2}{( 1 + x + 1 - x )} = 1 [p u tt in gx = 0]$

Example 8: Evaluate $lim_{x \to 1} \frac{( x ^{2} - 4 x + 3 )}{( x - 1 )}$

Solution:

We have,

$lim_{x \to 1} \frac{( x ^{2} - 4 x + 3 )}{( x - 1 )}$

$= lim_{x \to 1} \frac{( x - 1 ) ( x - 3 )}{( x - 1 )}$

$= lim_{x \to 1} (x - 3)$

=-2

Example 9: Evaluate $lim_{x \to 0} (\frac{e ^{3 x} - 1}{x})$

Solution:

We have,

$lim_{x \to 0} (\frac{e ^{3 x} - 1}{x})$

$= lim_{3 x \to 0} {(\frac{e ^{3 x} - 1}{3 x}) \times 3}$

$[∵ (x \to 0) \Rightarrow (3 x \to 0)]$

$= 3 \times lim_{y \to 0} (\frac{e ^{y} - 1}{y})$ , where y = 3x

= (3 × 1) = 3 $[⊠ lim_{y \to 0} (\frac{e ^{y} - 1}{y}) = 1]$

Example 10: Evaluate $lim_{x \to 0} (\frac{3 ^{x} - 2 ^{x}}{x})$

Solution:

We have,

$lim_{x \to 0} (\frac{3 ^{x} - 2 ^{x}}{x})$

$= lim_{x \to 0} {\frac{( 3 ^{x} - 1 ) - ( 2 ^{x} - 1 )}{x}}$

$lim_{x \to 0} (\frac{3 ^{x} - 1}{x}) - lim_{x \to 0} (\frac{2 ^{x} - 1}{x})$

= (log 3 – log 2)

$= lo g \frac{3}{2} [lim_{x \to 0} (\frac{a ^{x} - 1}{x}) = lo g a] .$

Example 11: Find the derivative of $f (x) = sin (x) \cdot cos (x) .$

Solution:

This is a product of two functions, so use the product rule:

$f^{'} (x) = \frac{d}{d x} [sin (x) \cdot cos (x)]$

$f^{'} (x) = sin (x) \cdot \frac{d}{d x} [cos (x)] + cos (x) \cdot \frac{d}{d x} [sin (x)]$

$f^{'} (x) = sin (x) \cdot (- sin (x)) + cos (x) \cdot cos (x)$

$f^{'} (x) = - sin^{2} (x) + cos^{2} (x)$

Example 12: Find the derivative of f(x) = e^3x.

Solution:

Given: f(x) = e^3x

Use the chain rule:

$f^{'} (x) = \frac{d}{d x} [e^{3 x}]$

$f^{'} (x) = e^{3 x} \cdot \frac{d}{d x} [3 x]$

$f^{'} (x) = 3 e^{3 x}$

Example 13: Find the derivative of $f (x) = ln (2 x^{2} + 3 x) .$

Solution:

Given: $f (x) = ln (2 x^{2} + 3 x)$

Apply the chain rule:

$f^{'} (x) = \frac{1}{2 x ^{2} + 3 x} \cdot \frac{d}{d x} [2 x^{2} + 3 x]$

$f^{'} (x) = \frac{1}{2 x ^{2} + 3 x} \cdot (4 x + 3)$

Simplify:

$f^{'} (x) = \frac{4 x + 3}{2 x ^{2} + 3 x}$

3.0Practice Questions of Limits and Derivatives

$lim_{x \to 3} (\frac{x ^{2} + 9}{x + 3})$
$lim_{x \to \frac{1}{2}} (\frac{4 x ^{2} - 1}{2 x - 1})$
$lim_{x \to 0} (\frac{e ^{4 x} - 1}{x})$
Find the derivative of f(x) = 7x⁴ – 3x² + 5x – 9.
Find the derivative of $g (x) = x + \frac{1}{x}$ .
Find the derivative of f(x) = sin(3x).
Find the derivative of g(x) = tan(x)·sec(x).
Find the derivative of $f (x) = e^{2 x} \cdot ln (x)$ .
Find the derivative of $g (x) = lo g_{10} (x^{2} + 1)$ .

Answers:

3
2
4
f’(x) = 28x³− 6x + 5
$g^{'} (x) = 1 - \frac{1}{x ^{2}}$
f’(x) = 3 cos (3x)
g’(x) = sec³ (x) + sec (x). tan² (x)
$f^{'} (x) = e^{2 x} (2 ln (x) + \frac{1}{x})$
$g^{'} (x) = \frac{2 x}{( x ^{2} + 1 ) l n ( 10 )}$

4.0Sample Questions on Limits and Derivatives

What are limits in mathematics?

Ans: A limit in mathematics describes the value that a function approaches as the input (or variable) approaches a certain point. For instance, if we say $lim_{x \to a} f (x) = L$ , it means that as x gets closer to a, the function f(x) gets closer to the value L.

How are limits and derivatives related?

Ans: The concept of a derivative is built on the idea of limits. The derivative of a function at a point is defined as the limit of the average rate of change of the function as the interval approaches zero. Mathematically, it's expressed as $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$ .

How do you find the derivative of a function using the limit definition?

Ans: To find the derivative of a function using the limit definition, you calculate:

$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$

This formula calculates the slope of the secant line as the interval h approaches zero, effectively giving the slope of the tangent line at point x.

Also Read:-

Alternative Hypothesis	Taylor Series	Kurtosis And Skewness
Spearmens Rank Correlation	Parametric Equations	Arithmetic And Geometric Sequence
Area Between Two Curves in Calculus	Derivative of Inverse Trignometric Functions	Applications of Determinants And Matrices

1.0Learning Outcomes

After completing these revision notes, you will be able to:

Understand the concept of limits and evaluate standard limits.
Distinguish between left-hand and right-hand limits.
Understand the basic idea of continuity.
Interpret derivatives as the rate of change and slope of a tangent.
Apply basic differentiation rules.
Solve introductory Calculus problems confidently.
Apply limit and derivative concepts to JEE Main, JEE Advanced, and CBSE-level questions.
Build a strong foundation for higher Calculus.

2.05-Minute Quick Revision

Must-Revise Concepts

Limits
Left-hand limit (LHL)
Right-hand limit (RHL)
Continuity
Derivative
Instantaneous rate of change
Slope of tangent
Differentiation
Standard limits
Basic derivative rules

Important Formula Recall

Definition of Derivative:
f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}
Standard Limits:
\lim_{x\to0}\frac{\sin x}{x}=1
\lim_{x\to0}\frac{\tan x}{x}=1
\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12
\lim_{x\to0}\frac{e^x-1}{x}=1
Derivatives:
\frac{d}{dx}(x^n)=nx^{n-1}
\frac{d}{dx}(c)=0
\frac{d}{dx}(x)=1

Important Properties

A limit exists only if the left-hand limit and right-hand limit are equal.
Continuity implies that the function has no break at the given point.
The derivative represents the slope of the tangent to the curve.
Differentiation measures the instantaneous rate of change.
Limits form the foundation for the definition of derivatives.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Standard Limits	Very High	Moderate
Evaluation of Limits	Very High	Moderate
Left-Hand & Right-Hand Limits	High	Moderate
Continuity	High	Moderate
Definition of Derivative	Very High	Moderate
Basic Differentiation	Very High	Moderate
Mixed Calculus Problems	Very High	Moderate–High

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Limits and Derivatives in JEE Main, JEE Advanced, and CBSE examinations.

Formula / Standard Result	Expression
Definition of Derivative	$(f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h})$
Derivative of Constant	$(\frac{d}{d x} (c) = 0)$
Derivative of (x)	$(\frac{d}{d x} (x) = 1)$
Power Rule	$(\frac{d}{d x} (x^{n}) = n x^{n - 1})$
Sum/Difference Rule	$(\frac{d}{d x} [f (x) \pm g (x)] = f^{'} (x) \pm g^{'} (x))$
Constant Multiple Rule	$(\frac{d}{d x} [c f (x)] = c f^{'} (x))$
Standard Limit	$(x \to 0 lim \frac{sin x}{x} = 1)$
Standard Limit	$(x \to 0 lim \frac{tan x}{x} = 1)$
Standard Limit	$(x \to 0 lim \frac{1 - cos x}{x ^{2}} = \frac{1}{2})$
Standard Limit	$(x \to 0 lim \frac{e ^{x} - 1}{x} = 1)$
Standard Limit	$(x \to 0 lim \frac{a ^{x} - 1}{x} = ln a,; a > 0)$

5.0Common Mistakes & JEE Tips

Limits and Derivatives is the gateway to Calculus and requires strong conceptual understanding. Most errors occur due to incorrect application of standard limits, improper algebraic simplification, or confusion while using the definition of the derivative.

Common Mistake	Correct Approach
Directly substituting values in indeterminate forms	Simplify the expression first using algebraic techniques or standard limits.
Confusing Left-Hand Limit (LHL) and Right-Hand Limit (RHL)	Check both limits separately before concluding whether the limit exists.
Forgetting standard limit formulas	Memorize the most frequently used limits for quick evaluation.
Using the derivative formula incorrectly	Apply the definition carefully by substituting (f(x+h)) correctly.
Errors in algebraic simplification	Factorize, rationalize, or cancel common terms before taking limits.
Missing negative signs during differentiation	Differentiate step by step and verify signs carefully.
Confusing continuity with differentiability	A function may be continuous but not differentiable at a point.
Ignoring domain restrictions	Ensure the function is defined near the point where the limit is evaluated.

6.0ALLEN Faculty Tips

Memorize all standard limit results thoroughly.
Practice algebraic simplification techniques such as factorization and rationalization.
Learn the derivative definition conceptually before memorizing formulas.
Understand the graphical meaning of limits and derivatives.
Revise basic differentiation rules regularly.
Focus on conceptual clarity instead of rote learning.
Solve Previous Years' Questions (PYQs) to understand the JEE examination pattern.

7.0PYQ Trend Analysis

Limits and Derivatives is one of the most important introductory Calculus chapters in JEE Main and JEE Advanced. Questions are commonly asked on standard limits, evaluation of limits, continuity, the definition of derivatives, and basic differentiation.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Standard Limits	2–3	Moderate
Evaluation of Limits	2–4	Moderate
Left-Hand & Right-Hand Limits	1–2	Moderate
Continuity	1–2	Moderate
Definition of Derivative	2–3	Moderate
Basic Differentiation	2–4	Moderate
Mixed Calculus Problems	2–4	Moderate–High

8.0Most Common Question Types

Evaluating limits using standard limit formulas.
Solving indeterminate forms through algebraic simplification.
Comparing Left-Hand Limit (LHL) and Right-Hand Limit (RHL).
Checking the continuity of a function at a point.
Finding derivatives using the first principle (definition).
Applying basic differentiation rules.
Solving introductory Calculus problems involving limits and derivatives.
JEE Advanced questions based on conceptual understanding of Calculus.

9.0Smart Revision Strategy

Follow this revision sequence to master the chapter efficiently:

Revise the concept of limits and standard limit formulas.
Practice evaluating limits using factorization, rationalization, and substitution techniques.
Understand the difference between LHL, RHL, and continuity.
Learn the definition of the derivative from first principles.
Memorize the basic rules of differentiation.
Solve introductory derivative problems.
Practice Previous Years' Questions (PYQs).
Finish with mixed Calculus problems under timed conditions.

Class 11 Maths Chapters List:- JEE & NEET Revision Notes

Chapter 1 Sets

Chapter 2 Relations and Functions

Chapter 3 Trigonometric Functions

Chapter 4 Complex Numbers and Quadratic Equations

Chapter 5 Linear Inequalities

Chapter 6 Permutations and Combinations

Chapter 7 Binomial Theorem

Chapter 8 Sequence and Series

Chapter 9 Straight Lines

Chapter 10 Conic Sections

Chapter 11 Introduction to Three-dimensional Geometry

Chapter 12 Limits and Derivatives

Chapter 13 Statistics

Chapter 14 Probability

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Class 11 Maths Chapter 12 Limits and Derivatives Revision Notes

The concepts of Limits and Derivatives mark the beginning of Calculus, one of the most important branches of Mathematics. Whether it is calculating the speed of a moving object, analyzing the growth of a population, optimizing engineering designs, or studying rates of change in Physics and Economics, derivatives play a crucial role. This chapter introduces the fundamental ideas of limits, the behavior of functions near a point, and the concept of the derivative as the instantaneous rate of change and slope of a tangent. These concepts form the foundation for advanced Calculus and are extensively tested in JEE Main, JEE Advanced, and CBSE Board examinations.

At ALLEN, our expert faculty have prepared these Class 11 Maths Chapter 12: Limits and Derivatives Revision Notes to help students build strong conceptual clarity in Calculus. These notes include concise theory, standard limit results, derivative formulas, shortcut techniques, solved illustrations, and previous years' question analysis. Whether you're preparing for board examinations or competitive exams, these revision notes will help you master the fundamentals of Calculus and solve problems with confidence.

1.0Chapter Snapshot

Topic	Overview
Limits	Concept and evaluation of limits
Standard Limits	Frequently used limit formulas
Left-Hand and Right-Hand Limits	Existence of limits
Continuity (Basic Idea)	Behavior of functions near a point
Derivative	Instantaneous rate of change
Geometrical Interpretation	Slope of a tangent
Algebra of Derivatives	Basic derivative rules
Applications	Foundation of Calculus and JEE problems

2.0Chapter 12 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution

Limits and Derivatives

Limits and Derivatives are fundamental concepts in calculus that lay the groundwork for more advanced topics. Understanding them is crucial for solving problems related to rates of change, optimization, and motion, among others.

1.0What are Limits and Derivatives in Mathematics?

Limits help us understand the behavior of functions as they approach a certain point. Formally, the limit of a function f(x) as x approaches a value a is the value that f(x) gets closer to as x gets closer to a. It’s expressed as:

$lim_{x \to a} f (x) = L$

Where L is the limit value.

Following fig. show the graphs of three functions. Note that in part(c), f(a) is not defined and part (b), f(a) ≠ L. But in each case, it happens at a. It is true that $lim_{x \to a} f (x) = L$ _.

Derivatives, on the other hand, measure how a function changes as its input changes. It's the rate at which the function's value changes with respect to change in the variable. The derivative of a function f(x) at a point x = a is given by:

$f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$

This formula is essentially the limit of the average rate of change as the interval shrinks to zero.

2.0Left hand Limits and Right Hand limit of a Function

$lim_{x \to a^{-}} f (x) = L$ $lim_{x \to a^{+}} f (x) = L$

$lim_{x \to a} f (x) = L$

If $lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = L (finite)$

Geometrical Meaning:

Tangent to the curve y = f(x) at a points (a, f(a)) is a limiting case of slope of secant through A.

$lim_{B \to A} m_{A B} = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$

$lim_{C \to A} m_{A c} = lim_{h \to 0} \frac{f ( a - h ) - f ( a )}{- h}$

Derivative exists if

$lim_{h \to 0} \frac{f ( a - h ) - f ( a )}{- h} = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$

3.0Limits and Derivatives Formulas

Algebra of Limits:

$lim_{x \to a} c = c$ , where c is a constant.
$lim_{x \to a} x = a$
$lim_{x \to a} [f (x) \pm g (x)] = lim_{x \to a} f (x) \pm lim_{x \to a} g (x)$
$lim_{x \to a} [f (x) \cdot g (x)] = lim_{x \to a} f (x) \cdot lim_{x \to a} g (x)$
$lim_{x \to a} \frac{f ( x )}{g ( x )} = \frac{l i m _{x \to a} f ( x )}{l i m _{x \to a} g ( x )}$
$lim_{x \to a} [(f) (x)] = lim_{x \to a} f (x)$

Limits of Some Basic Trigonometric Functions

$lim_{x \to 0} \frac{s i n x}{x} = 1$
$lim_{x \to 0} \frac{t a n x}{x} = 1$
$lim_{x \to 0} \frac{s i n ^{- 1} x}{x} = 1$
$lim_{x \to 0} \frac{t a n ^{- 1} x}{x} = 1$ (Where x is measured in radians)
If $lim_{x \to 0} f (x) = 0 t han lim_{x \to 0} \frac{s i n f ( x )}{f ( x )} = 1$ e.g. $lim_{x \to 1} \frac{s i n ( l n x )}{( l n x )} = 1$
$lim_{x \to 0} \frac{1 - c o s x}{x} = 0$

Limits of Some Exponential and Logarithmic Functions

$lim_{x \to 0} e^{x} = 1$
$lim_{x \to 0} \frac{e ^{x} - 1}{x} = 1$
$lim_{x \to \infty} (1 + \frac{1}{x})^{x} = e$
$lim_{x \to \infty} (1 + \frac{a}{x})^{x} = e^{a}$
$lim_{x \to 0} (1 + x)^{1/ x} = e$
$lim_{x \to 0} \frac{a ^{x} - 1}{x} = lo g_{e} a$
$lim_{x \to 0} \frac{l o g ( 1 + x )}{x} = 1$

xⁿFormula

$lim_{x \to a} \frac{x ^{n} - a ^{n}}{x - a} = n (a)^{n - 1}$

Basic Derivative Formulas:

$\frac{d}{d x} [c] = 0$ , where c is a constant.
$\frac{d}{d x} [x^{n}] = n x^{n - 1}$
$\frac{d}{d x} [sin x] = cos x$
$\frac{d}{d x} [cos x] = - sin x$

Algebra of Derivative of Functions:

$\frac{d}{d x} [f (x) \pm g (x)] = \frac{d}{d x} f (x) \pm \frac{d}{d x} g (x)$

$\frac{d}{d x} [f (x) \times g (x)] = f (x) \cdot \frac{d}{d x} g (x) + g (x) \cdot \frac{d}{d x} f (x)$

$\frac{d}{d x} [\frac{f ( x )}{g ( x )}] = \frac{g ( x ) \cdot \frac{d}{d x} f ( x ) - f ( x ) \cdot \frac{d}{d x} g ( x )}{( g ( x ) ) ^{2}}$

1.0Relationship Between Derivatives and Limits

The concept of derivatives is fundamentally built on the concept of limits. A derivative is essentially a limit that describes the rate of change of a function at a specific point. Without limits, the definition of a derivative wouldn't exist. When we calculate a derivative using the definition, we're actually finding the limit of the function's average rate of change as the interval over which the change is measured approaches zero.

2.0Solved Examples of Limits and Derivatives

Example 1: Evaluate the limit: $lim_{x \to 3} \frac{x ^{2} - 9}{x - 3}$

Solution:

Given: $lim_{x \to 3} \frac{x ^{2} - 9}{x - 3}$

$= lim_{x \to 3} \frac{x ^{2} - 9}{x - 3}$

Simplify the expression

$= lim_{x \to 3} \frac{( x - 3 ) ( x + 3 )}{x - 3}$

Cancelling the common factor (x – 3)

$= lim_{x \to 3} x + 3$

Now, substitute x = 3 into the simplified expression

= 3 + 3 = 6

So, $lim_{x \to 3} \frac{x ^{2} - 9}{x - 3} = 6$

Example 2: Find the derivative of f(x) = x² at x = 2.

Solution:

Using the definition of a derivative:

$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$

Substituting f(x) = x²:

$f^{'} (2) = lim_{h \to 0} \frac{( 2 + h ) ^{2} - 4}{h}$

$f^{'} (2) = lim_{h \to 0} \frac{4 + 4 h + h ^{2} - 4}{h}$

$f^{'} (2) = lim_{h \to 0} (4 + h)$

$f^{'} (2) = 4$

So, f'(2) = 4.

Example 3: Evaluate the limit: $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$

Solution:

Given: $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$

$= lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$

Simplify the expression

$= lim_{x \to 2} \frac{( x - 2 ) ( x + 2 )}{x - 2}$

Cancelling the common factor (x – 2)

$= lim_{x \to 2} x + 2$

Now, substitute x = 2 into the simplified expression

= 2 + 2 = 4

So, $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2} = 4$

Example 4: Find the derivative of f(x) = 3x³ – 5x² + 2x – 7.

Solution:

Using the power rule $\frac{d}{d x} [x^{n}] = n x^{n - 1}$ , differentiate each term:

$f^{'} (x) = 3 \cdot 3 x^{2} - 5 \cdot 2 x^{1} + 2 \cdot 1 x^{0} - 0$

$f^{'} (x) = 9 x^{2} - 10 x + 2$

Example 5: Find the derivative of f(x) = x² + 3x using the definition of a derivative.

Solution:

The derivative is given by:

$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$

Substitute f(x) = x² + 3x:

Expand and simplify:

$f^{'} (x) = lim_{h \to 0} \frac{( x + h ) ^{2} + 3 ( x + h ) - ( x ^{2} + 3 x )}{h}$

$f^{'} (x) = lim_{h \to 0} \frac{2 x h + h ^{2} + 3 h}{h}$

Factor out h:

$f^{'} (x) = lim_{h \to 0} (2 x + h + 3)$

$f^{'} (x) = 2 x + 3$

So, the derivative is f'(x) = 2x + 3.

Example 6: Evaluate the limit $lim_{x \to \infty} \frac{5 x ^{3} - 2 x ^{2} + 4}{3 x ^{3} + x - 7}$

Solution:

Given: $lim_{x \to \infty} \frac{5 x ^{3} - 2 x ^{2} + 4}{3 x ^{3} + x - 7}$

Divide each term by x³ (the highest power of x in the denominator)

$lim_{x \to \infty} \frac{5 - \frac{2}{x} + \frac{4}{x ^{3}}}{3 + \frac{1}{x ^{2}} - \frac{7}{x ^{3}}}$

Putting the limit x = infinite

$lim_{x \to \infty} \frac{5 - \frac{2}{x} + \frac{4}{x ^{3}}}{3 + \frac{1}{x ^{2}} - \frac{7}{x ^{3}}} = \frac{5}{3}$

So, $lim_{x \to \infty} \frac{5 x ^{3} - 2 x ^{2} + 4}{3 x ^{3} + x - 7} = \frac{5}{3}$

Example 7: Evaluate $lim_{x \to 0} {\frac{1 + x - 1 - x}{x}}$

Solution:

We have

$lim_{x \to 0} {\frac{1 + x - 1 - x}{x}}$

$= lim_{x \to 0} {\frac{1 + x - 1 - x}{x} \cdot \frac{( 1 + x + 1 - x )}{( 1 + x + 1 - x )}}$

$= lim_{x \to 0} \frac{{( 1 + x ) - ( 1 - x )}}{x ( 1 + x + 1 - x )}$

$= lim_{x \to 0} \frac{2 x}{x ( 1 + x + 1 - x )}$

$= lim_{x \to 0} \frac{2}{( 1 + x + 1 - x )} = 1 [p u tt in gx = 0]$

Example 8: Evaluate $lim_{x \to 1} \frac{( x ^{2} - 4 x + 3 )}{( x - 1 )}$

Solution:

We have,

$lim_{x \to 1} \frac{( x ^{2} - 4 x + 3 )}{( x - 1 )}$

$= lim_{x \to 1} \frac{( x - 1 ) ( x - 3 )}{( x - 1 )}$

$= lim_{x \to 1} (x - 3)$

=-2

Example 9: Evaluate $lim_{x \to 0} (\frac{e ^{3 x} - 1}{x})$

Solution:

We have,

$lim_{x \to 0} (\frac{e ^{3 x} - 1}{x})$

$= lim_{3 x \to 0} {(\frac{e ^{3 x} - 1}{3 x}) \times 3}$

$[∵ (x \to 0) \Rightarrow (3 x \to 0)]$

$= 3 \times lim_{y \to 0} (\frac{e ^{y} - 1}{y})$ , where y = 3x

= (3 × 1) = 3 $[⊠ lim_{y \to 0} (\frac{e ^{y} - 1}{y}) = 1]$

Example 10: Evaluate $lim_{x \to 0} (\frac{3 ^{x} - 2 ^{x}}{x})$

Solution:

We have,

$lim_{x \to 0} (\frac{3 ^{x} - 2 ^{x}}{x})$

$= lim_{x \to 0} {\frac{( 3 ^{x} - 1 ) - ( 2 ^{x} - 1 )}{x}}$

$lim_{x \to 0} (\frac{3 ^{x} - 1}{x}) - lim_{x \to 0} (\frac{2 ^{x} - 1}{x})$

= (log 3 – log 2)

$= lo g \frac{3}{2} [lim_{x \to 0} (\frac{a ^{x} - 1}{x}) = lo g a] .$

Example 11: Find the derivative of $f (x) = sin (x) \cdot cos (x) .$

Solution:

This is a product of two functions, so use the product rule:

$f^{'} (x) = \frac{d}{d x} [sin (x) \cdot cos (x)]$

$f^{'} (x) = sin (x) \cdot \frac{d}{d x} [cos (x)] + cos (x) \cdot \frac{d}{d x} [sin (x)]$

$f^{'} (x) = sin (x) \cdot (- sin (x)) + cos (x) \cdot cos (x)$

$f^{'} (x) = - sin^{2} (x) + cos^{2} (x)$

Example 12: Find the derivative of f(x) = e^3x.

Solution:

Given: f(x) = e^3x

Use the chain rule:

$f^{'} (x) = \frac{d}{d x} [e^{3 x}]$

$f^{'} (x) = e^{3 x} \cdot \frac{d}{d x} [3 x]$

$f^{'} (x) = 3 e^{3 x}$

Example 13: Find the derivative of $f (x) = ln (2 x^{2} + 3 x) .$

Solution:

Given: $f (x) = ln (2 x^{2} + 3 x)$

Apply the chain rule:

$f^{'} (x) = \frac{1}{2 x ^{2} + 3 x} \cdot \frac{d}{d x} [2 x^{2} + 3 x]$

$f^{'} (x) = \frac{1}{2 x ^{2} + 3 x} \cdot (4 x + 3)$

Simplify:

$f^{'} (x) = \frac{4 x + 3}{2 x ^{2} + 3 x}$

3.0Practice Questions of Limits and Derivatives

$lim_{x \to 3} (\frac{x ^{2} + 9}{x + 3})$
$lim_{x \to \frac{1}{2}} (\frac{4 x ^{2} - 1}{2 x - 1})$
$lim_{x \to 0} (\frac{e ^{4 x} - 1}{x})$
Find the derivative of f(x) = 7x⁴ – 3x² + 5x – 9.
Find the derivative of $g (x) = x + \frac{1}{x}$ .
Find the derivative of f(x) = sin(3x).
Find the derivative of g(x) = tan(x)·sec(x).
Find the derivative of $f (x) = e^{2 x} \cdot ln (x)$ .
Find the derivative of $g (x) = lo g_{10} (x^{2} + 1)$ .

Answers:

3
2
4
f’(x) = 28x³− 6x + 5
$g^{'} (x) = 1 - \frac{1}{x ^{2}}$
f’(x) = 3 cos (3x)
g’(x) = sec³ (x) + sec (x). tan² (x)
$f^{'} (x) = e^{2 x} (2 ln (x) + \frac{1}{x})$
$g^{'} (x) = \frac{2 x}{( x ^{2} + 1 ) l n ( 10 )}$

4.0Sample Questions on Limits and Derivatives

What are limits in mathematics?

Ans: A limit in mathematics describes the value that a function approaches as the input (or variable) approaches a certain point. For instance, if we say $lim_{x \to a} f (x) = L$ , it means that as x gets closer to a, the function f(x) gets closer to the value L.

How are limits and derivatives related?

Ans: The concept of a derivative is built on the idea of limits. The derivative of a function at a point is defined as the limit of the average rate of change of the function as the interval approaches zero. Mathematically, it's expressed as $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$ .

How do you find the derivative of a function using the limit definition?

Ans: To find the derivative of a function using the limit definition, you calculate:

$f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$

This formula calculates the slope of the secant line as the interval h approaches zero, effectively giving the slope of the tangent line at point x.

Also Read:-

Alternative Hypothesis	Taylor Series	Kurtosis And Skewness
Spearmens Rank Correlation	Parametric Equations	Arithmetic And Geometric Sequence
Area Between Two Curves in Calculus	Derivative of Inverse Trignometric Functions	Applications of Determinants And Matrices

1.0Learning Outcomes

After completing these revision notes, you will be able to:

Understand the concept of limits and evaluate standard limits.
Distinguish between left-hand and right-hand limits.
Understand the basic idea of continuity.
Interpret derivatives as the rate of change and slope of a tangent.
Apply basic differentiation rules.
Solve introductory Calculus problems confidently.
Apply limit and derivative concepts to JEE Main, JEE Advanced, and CBSE-level questions.
Build a strong foundation for higher Calculus.

2.05-Minute Quick Revision

Must-Revise Concepts

Limits
Left-hand limit (LHL)
Right-hand limit (RHL)
Continuity
Derivative
Instantaneous rate of change
Slope of tangent
Differentiation
Standard limits
Basic derivative rules

Important Formula Recall

Definition of Derivative:
f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}
Standard Limits:
\lim_{x\to0}\frac{\sin x}{x}=1
\lim_{x\to0}\frac{\tan x}{x}=1
\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12
\lim_{x\to0}\frac{e^x-1}{x}=1
Derivatives:
\frac{d}{dx}(x^n)=nx^{n-1}
\frac{d}{dx}(c)=0
\frac{d}{dx}(x)=1

Important Properties

A limit exists only if the left-hand limit and right-hand limit are equal.
Continuity implies that the function has no break at the given point.
The derivative represents the slope of the tangent to the curve.
Differentiation measures the instantaneous rate of change.
Limits form the foundation for the definition of derivatives.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Standard Limits	Very High	Moderate
Evaluation of Limits	Very High	Moderate
Left-Hand & Right-Hand Limits	High	Moderate
Continuity	High	Moderate
Definition of Derivative	Very High	Moderate
Basic Differentiation	Very High	Moderate
Mixed Calculus Problems	Very High	Moderate–High

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Limits and Derivatives in JEE Main, JEE Advanced, and CBSE examinations.

Formula / Standard Result	Expression
Definition of Derivative	$(f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h})$
Derivative of Constant	$(\frac{d}{d x} (c) = 0)$
Derivative of (x)	$(\frac{d}{d x} (x) = 1)$
Power Rule	$(\frac{d}{d x} (x^{n}) = n x^{n - 1})$
Sum/Difference Rule	$(\frac{d}{d x} [f (x) \pm g (x)] = f^{'} (x) \pm g^{'} (x))$
Constant Multiple Rule	$(\frac{d}{d x} [c f (x)] = c f^{'} (x))$
Standard Limit	$(x \to 0 lim \frac{sin x}{x} = 1)$
Standard Limit	$(x \to 0 lim \frac{tan x}{x} = 1)$
Standard Limit	$(x \to 0 lim \frac{1 - cos x}{x ^{2}} = \frac{1}{2})$
Standard Limit	$(x \to 0 lim \frac{e ^{x} - 1}{x} = 1)$
Standard Limit	$(x \to 0 lim \frac{a ^{x} - 1}{x} = ln a,; a > 0)$

5.0Common Mistakes & JEE Tips

Limits and Derivatives is the gateway to Calculus and requires strong conceptual understanding. Most errors occur due to incorrect application of standard limits, improper algebraic simplification, or confusion while using the definition of the derivative.

Common Mistake	Correct Approach
Directly substituting values in indeterminate forms	Simplify the expression first using algebraic techniques or standard limits.
Confusing Left-Hand Limit (LHL) and Right-Hand Limit (RHL)	Check both limits separately before concluding whether the limit exists.
Forgetting standard limit formulas	Memorize the most frequently used limits for quick evaluation.
Using the derivative formula incorrectly	Apply the definition carefully by substituting (f(x+h)) correctly.
Errors in algebraic simplification	Factorize, rationalize, or cancel common terms before taking limits.
Missing negative signs during differentiation	Differentiate step by step and verify signs carefully.
Confusing continuity with differentiability	A function may be continuous but not differentiable at a point.
Ignoring domain restrictions	Ensure the function is defined near the point where the limit is evaluated.

6.0ALLEN Faculty Tips

Memorize all standard limit results thoroughly.
Practice algebraic simplification techniques such as factorization and rationalization.
Learn the derivative definition conceptually before memorizing formulas.
Understand the graphical meaning of limits and derivatives.
Revise basic differentiation rules regularly.
Focus on conceptual clarity instead of rote learning.
Solve Previous Years' Questions (PYQs) to understand the JEE examination pattern.

7.0PYQ Trend Analysis

Limits and Derivatives is one of the most important introductory Calculus chapters in JEE Main and JEE Advanced. Questions are commonly asked on standard limits, evaluation of limits, continuity, the definition of derivatives, and basic differentiation.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Standard Limits	2–3	Moderate
Evaluation of Limits	2–4	Moderate
Left-Hand & Right-Hand Limits	1–2	Moderate
Continuity	1–2	Moderate
Definition of Derivative	2–3	Moderate
Basic Differentiation	2–4	Moderate
Mixed Calculus Problems	2–4	Moderate–High

8.0Most Common Question Types

Evaluating limits using standard limit formulas.
Solving indeterminate forms through algebraic simplification.
Comparing Left-Hand Limit (LHL) and Right-Hand Limit (RHL).
Checking the continuity of a function at a point.
Finding derivatives using the first principle (definition).
Applying basic differentiation rules.
Solving introductory Calculus problems involving limits and derivatives.
JEE Advanced questions based on conceptual understanding of Calculus.

9.0Smart Revision Strategy

Follow this revision sequence to master the chapter efficiently:

Revise the concept of limits and standard limit formulas.
Practice evaluating limits using factorization, rationalization, and substitution techniques.
Understand the difference between LHL, RHL, and continuity.
Learn the definition of the derivative from first principles.
Memorize the basic rules of differentiation.
Solve introductory derivative problems.
Practice Previous Years' Questions (PYQs).
Finish with mixed Calculus problems under timed conditions.

Frequently Asked Questions

What are the most important formulas to revise from Limits and Derivatives?

Which topics from Limits and Derivatives are frequently asked in JEE Main and JEE Advanced?

How many questions can you expect from Limits and Derivatives in JEE Main?

What are the most common mistakes students make while solving Limits and Derivatives questions?

What is the best last-minute revision strategy for Limits and Derivatives?

Join ALLEN!

Class 11 Maths Chapter 12 Limits and Derivatives Revision Notes

1.0Chapter Snapshot

2.0Chapter 12 Smart Study Material by ALLEN Experts: Download PDFs

On this page

Limits and Derivatives

1.0What are Limits and Derivatives in Mathematics?

2.0Left hand Limits and Right Hand limit of a Function

3.0Limits and Derivatives Formulas

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page

Frequently Asked Questions

What are the most important formulas to revise from Limits and Derivatives?

Which topics from Limits and Derivatives are frequently asked in JEE Main and JEE Advanced?

How many questions can you expect from Limits and Derivatives in JEE Main?

What are the most common mistakes students make while solving Limits and Derivatives questions?

What is the best last-minute revision strategy for Limits and Derivatives?

Join ALLEN!

Class 11 Maths Chapter 12 Limits and Derivatives Revision Notes

1.0Chapter Snapshot

2.0Chapter 12 Smart Study Material by ALLEN Experts: Download PDFs

On this page

Limits and Derivatives

1.0What are Limits and Derivatives in Mathematics?

2.0Left hand Limits and Right Hand limit of a Function

3.0Limits and Derivatives Formulas

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page