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 lefthand limit will be obtained when x = a – h or x > a^{–}
Similarly, the righthand 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

 \(\begin{array}{l}\underset{x\to 1}{\mathop{\lim }}\,\,{{x}^{2}}+1={{1}^{2}}+1=2\end{array} \)
 \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,{{x}^{2}}x={{0}^{2}}0=0\end{array} \)
 \(\begin{array}{l}\underset{x\to 2}{\mathop{\lim }}\,\,\frac{{{x}^{2}}4}{x+3}=\frac{44}{2+3}=0\end{array} \)
(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
 \(\begin{array}{l} \log (1+x)=x\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}\frac{{{x}^{4}}}{4}+…….\end{array} \)
 \(\begin{array}{l}{{e}^{x}}=1+x+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}\frac{{{x}^{4}}}{4!}+…….\end{array} \)
 \(\begin{array}{l}{{a}^{x}}=1+x\log a+\frac{{{x}^{2}}}{2!}{(log\;a)}^{2}+…….\end{array} \)
 \(\begin{array}{l}\sin x=x\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}……\end{array} \)
 \(\begin{array}{l}\cos x=1\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!}……\end{array} \)
 \(\begin{array}{l}\tan x=x+\frac{{{x}^{3}}}{3}+\frac{2}{15}{{x}^{5}}+……\end{array} \)
Some Important limits
 \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\sin x}{x}=1\end{array} \)
 \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{1\cos x}{{{x}^{2}}}=\frac{1}{2}\end{array} \)
 \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\tan x}{x}=1\end{array} \)
 \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{{{e}^{x}}1}{x}=1\end{array} \)
 \(\begin{array}{l}\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\log (1+x)}{x}=1\end{array} \)
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 (x^{3} – x)
(b)
Solution:
(a) f(x) = sgn (x^{3}–x)
Here, x^{3} – x = 0, so, x =0, –1, 1
Hence, f(x) is discontinuous at x = 0, 1, –1
(b)
Hence,
When 2/(1 + x^{2}) 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 x_{0} (a, b) for which f(x_{0}) = c
Limits, Continuity and Differentiability JEE Main Questions
12 MustDo 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 lefthand derivative
RHD: f’(0^{+}) =
LHD: F’
How can a function fail to be differentiable?
The function f(x) is said to be nondifferentiable 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
Frequently Asked Questions
Give the definition of continuous function in Mathematics.
A function is said to be continuous at a point x = a, if lim_{x→a} f(x) exists and lim_{x→a} f(x) = f(a).
What do you mean by differentiability?
A function f(x) is said to be differentiable at the point x = a if the derivative f’(a) exists at every point in its domain.
How do you check the continuity of a function f(x) at x = a?
We have to check the lefthand limit and righthand limit and the value of the function at a point x=a. If LHL = RHL = value of the function at x = a, then the function is continuous at x = a.
What do you mean by the second derivative of a function?
The second derivative is the derivative of a function. We denote the second derivative of f(x) by f’’(x).
Are continuity and differentiability related?
Yes, if a function is differentiable at any point in its domain, it will be continuous at that point; vice versa, it is not always true.
What is the value of lim_{x→ 0}(log (1+x)/x)?
The value of lim_{x→ 0}(log (1+x)/x) = 1.
What is the value of lim_{x→ 0 }(e^{x}1)/x?
The value of lim_{x→ 0 }(e^{x}1)/x = 1.
Is a continuous function always differentiable?
A continuous function need not be differentiable. If a function f(x) is differentiable at point a, then it is continuous at point a. But the converse is not true.
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