Limits

What are Limits? In Mathematics, limits are a fundamental concept in calculus that describes the behavior of functions as their input values approach certain points. Limits define the behavior of functions as their inputs approach specific values. Limits represent the value a function approaches as its input gets arbitrarily close to a certain point. Limits enable the precise analysis of functions' behavior and are crucial in calculus for defining continuity, derivatives, and integrals. Let’s elaborate more:

Consider function f(x), as x approaches a specific value c, denoted as ${\lim }_{x\to c}f(x)$, the limit represents the value that f(x) approaches as x gets arbitrarily close to c from both sides (left and right). 

For a limit to exist, the function's behavior must be consistent as x approaches c. This means that the values of f(x) must approach a single number, rather than diverging or oscillating wildly. 

Limits are crucial in analyzing the behavior of functions, determining continuity, and evaluating derivatives and integrals. Limits form the foundation of calculus and are essential in understanding various mathematical phenomena.

1.0Limits Definition 

When examining a real-valued function "f" and a real number "c," the limit is typically expressed as follows: 

${\lim }_{x\to c}f(x)=L$

"The limit of f(x) as x approaches c equals L." The notation "lim" denotes the limit, while the arrow signifies that the function f(x) converges to the limit L as x approaches c.

2.0Properties of Limits

Properties of limits are fundamental in analyzing functions and their behavior. Here are some key properties:

1. Linearity: The limit of a sum or difference of functions is the sum or difference of their limits.

• Sum Rule: ${\lim }_{x\to a}\{f(x)+g(x)\}={\lim }_{x\to a}f(x)+{\lim }_{x\to a}g(x)$

Example : ${\lim }_{x\to 2}\left(x^2+x\right)={\lim }_{x\to 2}{x}^2+{\lim }_{x\to 2}x,22+2=6$

• Difference Rule: ${\lim }_{x\to a}\{f(x)-g(x)\}={\lim }_{x\to a}f(x)-{\lim }_{x\to a}g(x)$

Example : ${\lim }_{x\to 2}\left(x^2-x\right)={\lim }_{x\to 2}{x}^2-{\lim }_{x\to 2}x=2^2-2=2$

2. Product Rule: The limit of a product of functions is the product of their limits.

${\lim }_{x\to a}f(x)\cdot g(x)={\lim }_{x\to a}f(x)\cdot {\lim }_{x\to a}g(x)$

Example : ${\lim }_{x\to 2}x(x-1)={\lim }_{x\to 2}{x}_x{\lim }_{x\to 2}x-1=2\times (2-1)=2$

3. Quotient Rule: The limit of a quotient of functions is the quotient of their limits, provided the limit of the denominator is not zero.

${\lim }_{x\to a}\frac{f(x)}{g(x)}=\frac{{\lim }_{x\to a}f(x)}{{\lim }_{x\to a}g(x)},Provided{\lim }_{\mathrm{x}\to \mathrm{a}}\mathrm{g}(\mathrm{x})\ne 0$

Example: ${\lim }_{x\to 2}\frac{x-1}{x}={\lim }_{x\to }(x-1)/{\lim }_{x\to }x=\frac{1}{2}$

4. Constant Rule: The limit of a constant times a function is the constant times the limit of the function.

${\lim }_{x\to a}kf(x)=k{\lim }_{x\to a}f(x)$, Where K is constant

Example: ${\lim }_{x\to 1}3(x-1)=3{\lim }_{x\to 1}(x-1)=3\times 0=0$

5. Power Rule: The limit of the power of a function is the power of the limit of the function. If m and n are integers than ${\lim }_{x\to a}[f(x){]}^{m/n}={\left[{\lim }_{x\to \infty }f(x)\right]}^{m/n}$ Provided  ${\left[{\lim }_{x\to \infty }f(x)\right]}^{m/n}$ is a real number.  
6. Limit of a Composite Function: The limit of a composite function is the limit of the outer function evaluated at the limit of the inner function, provided the outer function is continuous at that point.

These properties provide powerful tools for evaluating limits and understanding the behavior of functions, enabling precise analysis in calculus and related fields.

3.0Limits and Functions

When considering the behavior of a function near a certain limit, it's possible for the function to approach two different limits: one as the variable approaches from values larger than the limit, i.e. Right hand Limit and another as it approaches from values smaller than the limit i.e. Left Hand Limit . In such cases, while the overall limit may not be defined, the right-hand and left-hand limits do exist.

The right-hand limit of a function represents the value the function approaches as the variable approaches its limit from the positive side.

This can be expressed as:

${\lim }_{x\to c^{+}}f(x)$

Similarly, the left-hand limit of a function represents the value the function approaches as the variable approaches its limit from the negative side.

This can be expressed as:

${\lim }_{x\to c^{-}}f(x)$

These concepts provide insights into the behavior of functions near specific points and are essential for analyzing their continuity and discontinuity.

The existence of the limit of a function hinge on the equality of its left-hand and right-hand limits.

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

4.0Limits of Functions and Continuity

Limits and Continuity are intricately linked concepts in calculus. A function can exhibit continuity or discontinuity. Continuity implies that small changes in the input of a function result in small changes in the output.

In elementary calculus, the condition f(x) → λ as x → a signifies that the value f(x) can be brought arbitrarily close to λ by taking x sufficiently close to, but not equal to, a. This indicates that f(a) may be far from λ, and it's not necessary for f(a) to even be defined. A crucial result used in function derivation is: f'(a) of a given function f at a number a can be expressed as…

$f^{\prime }(a)={\lim }_{x\to a}\frac{f(x)-f(a)}{x-a}$

5.0Limits of Complex Functions

To ascertain the differentiability of functions of a complex variable, utilize the following formula:

The function f(z) is considered differentiable at z = z₀ if the expression 

${\lim }_{\Delta z\to 0}\frac{f\left(z_0+{\Delta }_z\right)-\left(z_0\right)}{\Delta z}$ exists.

Here, Δz = Δx + iΔy.

6.0Limits of Trigonometric Functions

1. ${\lim }_{x\to 0}\frac{\sin x}{x}=1$

2. ${\lim }_{x\to 0}\frac{\tan x}{x}=1$

3. ${\lim }_{x\to 0}\frac{{\sin }^{-1}x}{x}=1$

4. ${\lim }_{x\to 0}\frac{{\tan }^{-1}x}{x}=1$ (Where x is measured in radians)

5. If ${\lim }_{x\to 0}f(x)=0then{\lim }_{x\to 0}\frac{\sin f(x)}{f(x)}=1$

 e.g. ${\lim }_{x\to 1}\frac{\sin (\ln x)}{(\mathrm{lin}x)}=1$

6. ${\lim }_{x\to 0}\frac{1-\cos x}{x}=0$

7.0Limits of Exponential Functions

${\lim }_{x\to 0}\frac{a^x-1}{x}=\ln a(a>0)$ In particular ${\lim }_{x\to 0}\frac{e^x-1}{x}=1$.

In general, if ${\lim }_{x\to a}f(x)=0$, then x ${\lim }_{x\to a}f(x)=0$

Indeterminant form: The form of function whose values can’t be determined analytically. for example: $\frac{0}{0},\frac{\infty }{\infty },\infty -\infty$ etc.

8.0Solved Examples on Limits

Example 1: Find the limit of the function $f(x)=\frac{x^2-4}{x-2}$ as x approaches 2.

Solution: To find the limit as x approaches 2, we can try direct substitution.

$f(x)=\frac{x^2-4}{x-2}$

At x = 2,  f(2) = $\frac{x^2-4}{x-2}=\frac{0}{0}$

Here we get an indeterminate from $\frac{0}{0}$ indicating that we need to simplify the expression further we can factorize the numerator.

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

Now, we can cancel out the common factor of x – 2.

f(x) = x +2

Now, let’s find the limit as a approaches ${\lim }_{x\to 2}f(x){\lim }_{x\to 2}(x+2)=2+2=4$

So, the limit of the function $\underset{x\to 2}{f(x)}=\frac{x^2-4}{x-2}$ is 4.

Example 2: Find the limit of the function $g(x)=\frac{\sin (x)}{x}$ as x approaches 0 

Solution: $\underset{x\to 0}{g(x)}=\frac{\sin (x)}{x}$

At x = 0, g(0)=$\frac{\sin (0)}{0}=\frac{0}{0}$

Here, we get an indeterminate form $\frac{0}{0}$

We can use the concept of limit of Trigonometric Functions ${\lim }_{x\to 0}\frac{\sin (x)}{x}=1$

Therefore, the limit of the function $g(x)=\frac{\sin x}{x}$ is 1.

Example 3: Find the limit of the Function $f(x)=\frac{x^3-8}{x-2}$ as x approaches 2

Solution: $\underset{x\to 0}{f(x)}=\frac{x^3-8}{x-2}$

At x = 2, $\underset{x\to 0}{f(x)}=\frac{x^3-8}{x-2}$

Applying factorization:

$f(x)=\frac{(x-2)\left(x^2+2x+4\right)}{(x-2)}$

Cancelling the common factor:

f(x) = x² + 2x + 4

Taking the limit as x approaches 2:

${\lim }_{x\to 2}f(x)={\lim }_{x\to 2}\left(x^2+2x+4\right)=2^2+2(2)+4=12$

Example 4: Find the limit of the function $g(x)\frac{{\sin }^2(x)}{x^2}$ as x approaches 0

Solution: $g(x)=\frac{{\sin }^2(x)}{x^2}$

At x = 0,  $g(0)=\frac{{\sin }^2(0)}{0^2}$

Using trigonometric limits:

${\lim }_{x\to \infty }\frac{{\sin }^2(x)}{x^2}=1$

Example 5: Find the limit of the function $h(x)=\frac{x^2-4x+4}{x-2}$ as x approaches 2.

Solution: $h(x)=\frac{x^2-4x+4}{x-2}$

At x = 2,  

$h(2)=\frac{2^2-4(2)+4}{2-2}=\frac{0}{0}$

Using factorization: $h(x)=\frac{(x-2{)}^2}{x-2}$

Cancelling the common factor:

h(x) = x – 2

Taking the limit as x approaches 2: 

${\lim }_{x\to 2}h(x)={\lim }_{x\to 2}(x-2)=2-2=0$

So, ${\lim }_{x\to 2}\frac{x^2-4x+4}{x-2}=0$

Example 6: Find the limit of the function $f(x)=\frac{\sqrt{x+1}-1}{x}$ as x approaches O.

Solution: $f(x)=\frac{\sqrt{x+1}-1}{x}$

At x = 0, $f(0)=\frac{\sqrt{0+1}-1}{x}=\frac{0}{0}$

Rationalize the numerator:

$f(x)=\frac{\sqrt{x+1}-1}{x}\times \frac{\sqrt{x+1}+1}{\sqrt{x+1}+1}$

$f(x)=\frac{x+1-1}{x(\sqrt{x+1}+1)}=\frac{x}{x(\sqrt{x+1}+1)}$

$f(x)=\frac{1}{\sqrt{x+1}+1}$

Taking the limit as x approaches 0:

${\lim }_{x\to x}f(x)=\frac{1}{\sqrt{0+1}+1}=\frac{1}{1+1}=\frac{1}{2}$

Example 7: Find the limit of the function $g(x)=\frac{\sin (3x)}{\tan (4x)}$ as x approaches $\frac{\pi }{12}$

Solution: $g(x)=\frac{\sin (3x)}{\tan (4x)}$

At x = $\frac{\pi }{12}$,

$g\left(\frac{\pi }{12}\right)=\frac{\sin \left(\frac{3\pi }{12}\right)}{\tan \left(\frac{4\pi }{12}\right)}=\frac{\sin \left(\frac{\pi }{4}\right)}{\tan \left(\frac{\pi }{3}\right)}$

Using trigonometric identities:

$\sin \left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}$

$\tan \left(\frac{\pi }{3}\right)-\sqrt{3}$

So, $g\left(\frac{\pi }{12}\right)=\frac{\frac{1}{\sqrt{2}}}{\sqrt{3}}=\frac{1}{\sqrt{6}}$

Example 8: Find the limit of the function ${\lim }_{x\to 0}\frac{e^{2x}-1}{x}$

Solution: ${\lim }_{x\to 0}\frac{e^{2x}-1}{2x}\times 2$

$=2{\lim }_{x\to 0}\frac{e^{2x}-1}{2x}=2\times 1=2$

9.0Sample Questions on Limits

1. What are some common indeterminate forms when finding limits?

Ans: Common indeterminate forms include $\frac{0}{0},\frac{\infty }{\infty },$ 0 × ∞, ∞ – ∞, 0°, 1^∞, and ∞°.

2. What is L'Hopital's Rule, and when is it used?

Ans: L'Hopital's Rule states that for certain indeterminate forms, the limit of the ratio of two functions is equal to the limit of the ratio of their derivatives. It is used when evaluating limits of the form $\frac{0}{0}$ or $\frac{\infty }{\infty }$.