Limits describe the value that a function approaches as the input approaches a certain point. Continuity refers to a function being smooth and unbroken at a specific point, meaning the function's limit at that point equals its actual value.
There are 2 main types of discontinuities: Non- Removable Discontinuity Jump Discontinuity: The function jumps from one value to another at a point. Infinite Discontinuity: The function approaches infinity near a point, causing a vertical asymptote. Removable Discontinuity: There is a hole in the function at a point where the limit exists, but the function is not defined or is defined differently.
imits focus on the behavior of a function as it approaches a specific point, describing what value the function gets closer to. Continuity deals with whether a function is smooth and unbroken at that point, ensuring that the limit at that point equals the function's actual value.
If a function is not continuous at a point, it has a discontinuity there. This could mean the function has a jump, an infinite behavior, or a removable discontinuity (like a hole). Such discontinuities can significantly affect the function's graph and its applicability in various contexts.
Yes, a function can have a limit at a point where it is not continuous. For example, a function with a removable discontinuity has a limit at the discontinuous point. The limit exists, but the function does not meet the continuity criteria because the function's value at the point does not match the limit.
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Limits and Continuity
Limits and Continuity are two of the most crucial concepts. These ideas form the foundation for understanding more advanced topics like derivatives and integrals. To grasp the essence of calculus, one must first understand the definition of limits and continuity and how they interrelate.
1.0What Are Limits and Continuity?
The concept of a Limit is fundamental to the study of calculus. In simple terms, the limit of a function at a particular point refers to the value that the function approaches as the input (or variable) approaches a specific value. It allows us to analyze how functions behave as they get close to a certain point, even if they never actually reach that point.
Continuity, on the other hand, is a property of a function that ensures it behaves in a predictable manner. A function is continuous if, roughly speaking, you can sketch its graph without lifting your pen. More formally, a function is considered continuous at a point if the limit of the function exists as it approaches that point is equal to the function’s value at that point.
2.0Definition of Limits and Continuity
Limit Definition: The limit of a function f(x) as x approaches a value c is the value that f(x) gets closer to as x gets closer to c. Mathematically, this is written as:
limx→cf(x)=L
This means that as x gets arbitrarily close to c, f(x) approaches L.
Continuity Definition: A function f(x) is continuous at a point x = c if the following three conditions are met:
f(c) is defined.
limx→cf(x)exists.
limx→cf(x)=f(c)
If a function meets these criteria at every point in its domain, it is considered continuous on that domain.
3.0The Difference Between Limits and Continuity
While limits deal with the behavior of functions as they approach a specific point, continuity concerns whether the function's graph is unbroken at that point. Simply put, limits are about approaching a value, whereas continuity is about the function actually attaining that value smoothly. If a function is continuous at a point, then the limit at that point equals the function's value there.
4.0Types of Discontinuity
Discontinuities occur when a function is not continuous at a point. There are several types of discontinuities:
Non Removable Discontinuity
Jump Discontinuity: The function has two different limit values as it approaches the point from the left and the right.
Infinite Discontinuity: The function increases or decreases without bound as it approaches the point.
Removable Discontinuity: The function has a "hole" at a point where the limit exists, but the function is not defined or is defined differently.
5.0Solved Examples of Limits and Continuity
Example 1: Find the limit of limx→1x−1x2−1 .
Solution:
If we substitute x = 1 into the function gives an indeterminate form 00.
=limx→1x−1(x−1)(x+1)(Splitting the numerator)
=limx→1(x+1)
= 1 + 1 = 2
So, limx→1x−1x2−1=2
Example 2: Evaluate: limx→∞3x2+2x−5x2+x+1
Solution:
limx→∞3x2+2x−5x2+x+1,(∞∞ form )
Put x=y1
=limy→03+2y−5y21+y+y2=31
Example 3: If limx→∞(x2+1x3+1−(ax+b))=2 , then
(A) a = 1, b = 1 (B) a = 1, b = 2(C) a = 1, b = –2(D) none of these
Ans. (C)
Solution:
limx→∞(x2+1x3+1−(ax+b))=2
⇒limx→∞x2+1x3(1−a)−bx2−ax+(1−b)=2
⇒limx→∞1+x21x(1−a)−b−xa+x2(1−b)=2
⇒ 1 – a = 0, –b = 2
⇒ a = 1, b = –2
Example 4: Evaluate: limx→01−cosxx3cotx
Solution:
limx→0sinx(1−cosx)x3cosx
=limx→0sinx⋅sin2xx3cosx(1+cosx)
=limx→0sin3xx3⋅cosx(1+cosx)=2
Example 5: If f(x)={sin2πx,[x]x<1x≥1 then find whether f(x) is continuous or not at x=1, where [.] denotes greatest integer function.
Solution:
f(x)={sin2πx,x<1[x],x≥1
For continuity at x=1, we determine, f(1), limx→1−f(x) and limx→1+f(x).
Now, f(1) = [1]=1
f(x)limx→1−=limx→1−sin2πx=sin2π=1 and limx→1+f(x)=limx→1+[x]=1
so f(1)=limx→1−f(x)=limx→1+f(x)
∴ f(x) is continuous at x = 1
Example 6: Discuss the continuity of f(x)=⎩⎨⎧∣x+1∣2x+3,x2+3,x3−15,,x<−2−2≤x<00≤x<3x≥3
Solution:
We write f(x) as f(x)=⎩⎨⎧−x−1,2x+3,x2+3,x3−15,x<−2−2≤x<00≤x<3x≥3
As we can see, f(x) is defined as a polynomial function in each of intervals (–∞, –2), [–2, 0), [0, 3) and [3, ∞). Therefore, the function is continuous in each of these four open intervals. Thus, we check the continuity at x = –2, 0, 3.
At the point x = –2
limx→−2−f(x)=limx→−2−(−x−1)
= +2 – 1
= 1
limx→−2+f(x)=limx→−2+(2x+3)
= 2·(–2) + 3 = –1
Therefore, limx→−2 f(x) does not exist and hence f(x) is discontinuous at x = –2.
At the point x = 0
limx→0−f(x)=limx→0−(2x+3)=3
limx→0+f(x)=limx→0+(x2+3)=3
f(0) = 02 + 3 = 3
Therefore f(x) is continuous at x=0.
At the point x = 3
limx→3−f(x)=limx→3−(x2+3)
=32 + 3 = 12
limx→3+f(x)=limx→3+(x3−15)
=33 – 15 = 12
f(3) =33 – 15 = 12
Therefore, f(x) is continuous at x = 3.
We find that f(x) is continuous at all points in R except at x = –2 .
6.0Practice Questions on Limits and Continuity
1.limx→12x2−7x+5x−1 is equal to
(A) 31(B) −31(C) 41(D) −41
2.limx→2x2−6x+8x3−6x2+11x−6 is equal to
(A) 23 (B) 21 (C) 1(D) Does not exist
3.limx→3x4−5x3+27x−27x3−7x2+15x−9 is equal to
(A) 2/9(B) 1/9(C) –2/9(D) Does not exist
4.The function f(x)=4x−x34−x2 , is -
(A) discontinuous at only one point in its domain.
(B) discontinuous at two points in its domain.
(C) discontinuous at three points in its domain.
(D) continuous everywhere in its domain.
5.A function y=f(x) is defined as f(x)={ksin6(x+3)πx−23−11−x for for x≤2x>2 . If f(x) is continuous at x=2, then find the value of 6k.
Question
1
2
3
4
5
Answer
C
B
A
D
2
7.0Sample Questions on Limits and Continuity
What is the Formal Definition of a Limit?
Ans: The limit of a function f(x) as x approaches a value c is the value L that f(x) gets closer to as x gets closer to c. Formally, this is expressed as:
limx→cf(x)=L
This indicates that as x nears c, f(x) approaches L, regardless of whether f(c) is defined.
What is the Meaning of Continuity?
Ans: A function is continuous at a point x = c if it is unbroken at that point. For a function to be continuous at c, the following must hold:
f(c) is defined.
limx→cf(x)exists.
limx→cf(x)=f(c).
This means the function has no jumps, holes, or vertical asymptotes at x = c .
