Limits, Continuity and Differentiability - Evaluations and Examples

The limit concept is certainly indispensable for the development of analysis, for convergence and divergence of infinite series also depend on this concept. The theory of limits and then defining continuity, differentiability and the definite integral in terms of the limit concept is successfully executed by mathematicians. In this section, you will study this concept in detail with the help of solved examples.

What Are Limits?

The expression

The neighbourhood of a point:

Let ‘a’ be real number and ‘h; is very close to ‘O’, then

The left-hand limit will be obtained when x = a – h or x -> a^–

Similarly, the right-hand limit will be obtained when x = a + h or x ->  a⁺

Related Concepts:

• Functions
• Limits, Continuity and Differentiability
• Differentiation
• Applications of Derivatives

Existence of Limit

The limit will exist if the following conditions get fulfilled:

(a)

(b) Both LHL & RHL should be finite.

Examples

• • \(\begin{array}{l}\underset{x\to 1}{\mathop{\lim }}\,\,{{x}^{2}}+1={{1}^{2}}+1=2\end{array} \)
  • \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,{{x}^{2}}-x={{0}^{2}}-0=0\end{array} \)
  • \(\begin{array}{l}\underset{x\to 2}{\mathop{\lim }}\,\,\frac{{{x}^{2}}-4}{x+3}=\frac{4-4}{2+3}=0\end{array} \)

(c) In limits, we have in determinant forms such as

In these cases, we try to simplify the problem into a valid function.

Watch This Video for More Reference

Calculus Problems

Expected Questions and Solutions

Use of Expansion in Evaluating Limits

Some Important Expansions

1. \(\begin{array}{l} \log (1+x)=x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}-\frac{{{x}^{4}}}{4}+…….\end{array} \)
2. \(\begin{array}{l}{{e}^{x}}=1+x+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}-\frac{{{x}^{4}}}{4!}+…….\end{array} \)
3. \(\begin{array}{l}{{a}^{x}}=1+x\log a+\frac{{{x}^{2}}}{2!}{(log\;a)}^{2}+…….\end{array} \)
4. \(\begin{array}{l}\sin x=x-\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}……\end{array} \)
5. \(\begin{array}{l}\cos x=1-\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!}……\end{array} \)
6. \(\begin{array}{l}\tan x=x+\frac{{{x}^{3}}}{3}+\frac{2}{15}{{x}^{5}}+……\end{array} \)

Some Important limits

1. \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\sin x}{x}=1\end{array} \)
2. \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{1-\cos x}{{{x}^{2}}}=\frac{1}{2}\end{array} \)
3. \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\tan x}{x}=1\end{array} \)
4. \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{{{e}^{x}}-1}{x}=1\end{array} \)
5. \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\log (1+x)}{x}=1\end{array} \)

Example: Solve

Solution:

Evaluation of Algebraic Limits

Direct substitution method

Example 1:

Example 2:

Factorisation method

Example:

Solution:

Rationalisation method

Example:

Solution:

Using Result:

Example:

Continuity

What Is Continuity?

A continuous function is a function for which small changes in the input results in small changes in the output. Otherwise, a function is said to be discontinuous.

A function f(x) is said to be continuous at x = a if

i.e., LHL = RHL = value of the function at x = a

Else, a function f(x) is said to be a discontinuous function.

Example 1:

Solution: Clearly, the function will be not defined for

Function is discontinuous for

Example 2: What value must be assigned to K so that the function

Solution:

Example 3: Discuss the continuity of

(a) Sgn (x³ – x)

(b) 

Solution:
(a) f(x) = sgn (x³–x)

Here, x³ – x = 0, so, x =0, –1, 1

Hence, f(x) is discontinuous at x = 0, 1, –1

(b)

Hence,

When 2/(1 + x²) is on integer

Example 4: Discuss the continuity of 

Solution:

Intermediate Value Theorem

If f is continuous on [a, b] and f(a) ≠ f(b), then for any value c ∈ (f(a), f(b)), there is at least one number in x₀ (a, b) for which f(x₀) = c

Limits, Continuity and Differentiability JEE Main Questions

12 Must-Do JEE Questions of Limits Continuity and Differentiability

Differentiability

A function, say f(x), is said to be differentiable at the point x = a if the derivative f ‘(a) exists at every point in its domain.

Existence of Derivative

Right and left-hand derivative

RHD: f’(0⁺) =

LHD: F’

How can a function fail to be differentiable?

The function f(x) is said to be non-differentiable at x = a, if

(a) Both RHD & LHD exist but are not equal

(b) Either or both RHD & LHD are not finite

(c) Either or both RHD & LHD do not exist.

Video Lesson:

Limits, Continuity and Differentiability – Part 1

Limits, Continuity and Differentiability – Part 2

Limits, Continuity and Differentiability – Important Topics

Limits, Continuity and Differentiability – Important Questions

Limits, Continuity and Differentiability – Revision Lesson Part 1

Limits, Continuity and Differentiability – Revision Lesson Part 2

Continuity of a Function

Theorems on Continuity

Differentiability and Its Conditions

Differentiability in an Interval