Mathematics · JEE

Important Questions: Limits, continuity and differentiability for JEE

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The most important Limits, continuity and differentiability questions for JEE cover conceptual traps, standard results, and numerical patterns from Limits, continuity and differentiability. Goodmarks provides 31+ high-yield MCQs with full solutions.

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Syllabus context

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

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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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What makes a Limits, continuity and differentiability question "important" for JEE?

Important questions test core concepts that appear frequently across JEE Main and Advanced papers — definitions, standard formulas, and classic problem types.

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Key areas include Limits, continuity and differentiability. Prioritise these before moving to edge cases.

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Revise 30–50 important MCQs per unit in the last month before JEE. Focus on questions you got wrong at least once.

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