Mathematics · JEE

Limits, continuity and differentiability Concepts for JEE

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Concept explainer

Limits, continuity and differentiability is a core JEE Main Mathematics subtopic under Limit, Continuity and Differentiability. Master the definitions, standard results, and typical MCQ patterns tested in JEE Main and Advanced.

Key points

Understand the definition and scope of Limits, continuity and differentiability in the JEE syllabus
Memorise key formulas and standard results linked to Limits, continuity and differentiability
Practise 20–40 syllabus-aligned MCQs with step-by-step solutions

JEE tips

Revise Limits, continuity and differentiability with a one-page formula sheet before attempting mixed tests
After each practice set, log mistakes specific to Limits, continuity and differentiability and reattempt after 48 hours

Common trap

Students often rush Limits, continuity and differentiability questions without checking units, sign conventions, or boundary conditions — always verify assumptions before calculating.

Syllabus context

Part of Limit, Continuity and Differentiability in JEE Main Mathematics.

Free sample questions

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Q1MathsUnit 7: Limit, Continuity and Differentiability

Let

\boldsymbol{f}(\boldsymbol{x})=\boldsymbol{x}^{3}-\boldsymbol{x}^{2}+\boldsymbol{x}+\mathbf{1}

and

\boldsymbol{g}(\boldsymbol{x})=

\left\{\begin{array}{l}\max \{f(t)\}, \quad 0 \leq t \leq x \quad 0 \leq x \leq 1 \\ 3-x, \quad 1<x \leq 2\end{array}\right.

Then in the interval

[0,2], g(x)

is This question has multiple correct options

Q2MathsUnit 7: Limit, Continuity and Differentiability

A polynomial

p(x)

when divided by

x^{2}-

3 x+2

leaves remainder

2 x-3 .

Then

Q3MathsUnit 7: Limit, Continuity and Differentiability

\operatorname{Let} f(x)=\left\{\begin{array}{cc}-1, & -2 \leq x<0 \\ x^{2}-1, & 0<x \leq 2\end{array}

and \right.

\boldsymbol{g}(\boldsymbol{x})=|\boldsymbol{f}(\boldsymbol{x})|+\boldsymbol{f}|\boldsymbol{x}|

then the number of points which

g(x)

is non differentiable, is

Q4MathsUnit 7: Limit, Continuity and Differentiability

Arrange the following limits in the ascending order: (1)

\lim _{x \rightarrow \infty}\left(\frac{1+x}{2+x}\right)^{x+2}

(2)

\lim _{x \rightarrow 0}(1+2 x)^{3 / x}

(3)

\lim _{\boldsymbol{\theta} \rightarrow \mathbf{0}} \frac{\sin \boldsymbol{\theta}}{\mathbf{2} \boldsymbol{\theta}}

(4)

\lim _{x \rightarrow 0} \frac{\log _{e}(1+x)}{x}

Q5MathsUnit 7: Limit, Continuity and Differentiability

Assertion

\lim _{\boldsymbol{x} \rightarrow \mathbf{0}} \frac{\sqrt{1-\cos 2 x}}{\boldsymbol{x}}

does not exist. Reason

|\sin x|=\left\{\begin{array}{cc}\sin x ; & 0<x<\frac{\pi}{2} \\ -\sin x ; & -\frac{\pi}{2}<x<0\end{array}\right.

Q6MathsUnit 7: Limit, Continuity and Differentiability

Let

\boldsymbol{f}(\boldsymbol{x})

be defined in the interval [-2,2] such that

f(x)=

\left\{\begin{array}{ll}-1, & -2 \leq x \leq 0 \\ x-1, & 0<x \leq 2\end{array} \text { and } g(x)=\right.

\boldsymbol{f}(|\boldsymbol{x}|)+|\boldsymbol{f}(\boldsymbol{x})|

Test the differentiablity of

g(x)

in (-2,2)

Q7MathsUnit 7: Limit, Continuity and Differentiability

The function

f(x)=

\left\{\begin{array}{l}\frac{\cos 3 x-\cos 4 x}{x^{2}}, \text { for } x \neq 0 \\ \frac{7}{2}, \text { for } x=0\end{array}

at \right.

\boldsymbol{x}=\mathbf{0}

Q8MathsUnit 7: Limit, Continuity and Differentiability

Assertion If

f(x)=0

has two distinct positive real roots then number of non- differentiable points of

\boldsymbol{y}=|\boldsymbol{f}(-|\boldsymbol{x}|)|

\mathbf{1}

Reason Graph of

\boldsymbol{y}=\boldsymbol{f}(|\boldsymbol{x}|)

is symmetrical about y-axis

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