$\mathop {\lim }\limits_{x \to 0} \frac{1}{x}\left\{ {\mathop \smallint \limits_y^a {e^{{{\sin }^2}t}}dt - \mathop \smallint \limits_{x + y}^a {e^{{{\sin }^2}t}}dt} \right\}$ is equal to (where $a$ is a constant)

$\sin \,2y\,\,{e^{{{\sin }^2}y}}$

Back to LIMITS

Your Progress

0 / 930 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • LIMITS

Limits practice · Mixed difficulty

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

Take a chapter quiz →

Showing 661-690 of 930 questions

Preparing practice and saved progress…

661 Easy

661 of 930

Application +4 | -1

The value of

\lim\limits_{x \to \infty } {\left( {\frac{{x + 3}}{{x + 1}}} \right)^{x + 2}}

View answer and explanation

Correct option: C

{e^2}

Option C (

{e^2}

) is correct because the limit can be rewritten in the form

\lim_{x \to \infty} (1 + \frac{a}{x})^x = e^a

662 Medium

662 of 930

Analyzing +4 | -1

Evaluate

\lim \limits_{{x \rightarrow \infty}}\left(\frac{x^3}{3 x^2-4}-\frac{x^2}{3 x+2}\right)

View answer and explanation

Correct option: B

\frac{2}{9}

Option B (

\frac{2}{9}

) is correct because after simplifying the expression and taking the limit as x approaches infinity, the result is

\frac{2}{9}

663 Easy

663 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow-\pi}} \frac{|x+\pi|}{\sin x}

is equal to

View answer and explanation

Correct option: D

Does not exist

Option D (Does not exist) is correct because the left-hand limit and right-hand limit are not equal.

664 Easy

664 of 930

Application +4 | -1

\lim\limits_{x \to 0} \frac{{\log \left( {3 + x} \right) - \log \left( {3 - x} \right)}}{x} = k

, the value of

k

View answer and explanation

Correct option: C

\frac{2}{3}

Option C (

\frac{2}{3}

) is correct because applying L'Hopital's rule to the limit results in

\frac{2}{3}

665 Easy

665 of 930

Analyzing +4 | -1

\lim\limits_{x \to \frac{\pi }{2}} \frac{{\left[ {1 - \tan \left( {\frac{x}{2}} \right)} \right]\left( {1 - \sin x} \right)}}{{\left[ {1 + \tan \left( {\frac{x}{2}} \right)} \right]{{\left( {\pi - 2x} \right)}^3}}}

View answer and explanation

Correct option: C

\frac{1}{{32}}

Option C (

\frac{1}{{32}}

) is correct because, by substituting

x = \frac{\pi}{2} - h

and using trigonometric identities and limits, the expression simplifies to 1/32.

666 Easy

666 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 0}} \frac{\tan x-\sin x}{x^3}

, is

View answer and explanation

Correct option: A

\frac{1}{2}

The correct answer is A (1/2) because by rewriting the expression and using trigonometric identities and limits, the value is found to be 1/2.

667 Medium

667 of 930

Analyzing +4 | -1

\lim _{x \rightarrow 0}(\cos x)^{\cot x}=

View answer and explanation

Correct option: A

Option A (1) is correct because by converting the expression to exponential form and using standard limits, the result simplifies to 1.

668 Easy

668 of 930

Application +4 | -1

\lim\limits_{x \to 0} \frac{{{{\left( {1 + x} \right)}^8} - 1}}{{{{\left( {1 + x} \right)}^2} - 1}}

is equal to

View answer and explanation

Correct option: C

Option C (4) is correct because after factoring and simplifying the expression, the limit as x approaches 0 is 4.

669 Easy

669 of 930

Application +4 | -1

The value of

\lim \limits_{{x \rightarrow-2}} \frac{\left(x^2-x-6\right)^2}{(x+2)^2}

, is

View answer and explanation

Correct option: B

Option B (25) is correct because it is the result of factoring and simplifying the limit expression as x approaches -2.

670 Easy

670 of 930

Application +4 | -1

\operatorname{Lim}_{x \rightarrow 0} \frac{\ln (3+x)-\ln (3-x)}{x}=k

, the value of

k

View answer and explanation

Correct option: A

\frac{2}{3}

Option A (

\frac{2}{3}

) is correct because it is the limit of the given expression as x approaches 0, evaluated using L'Hôpital's rule or series expansion.

671 Medium

671 of 930

Application +4 | -1

Which of the following is/are true?

I. The derivative of

-x

-1.

II. The derivative of

(-{x})^{-1}

\frac{1}{{x}^2}

View answer and explanation

Correct option: A

Both I and II are true

Option A is correct because both statements are true. The derivative of -x is -1, and the derivative of (-x)^(-1) is 1/x².

672 Medium

672 of 930

Analyzing +4 | -1

Let

f(x)=\frac{\tan x}{x}

then

{\log _e}\left( {\mathop {\operatorname{Lt} }\limits_{x \to 0} {{\left( {[f(x)] + {x^2}} \right)}^{\frac{1}{{\{ f(x)\} }}}}} \right) =

________ Where [.] is G.I.F

\{\cdot\}

is fractional part function

View answer and explanation

Correct option: C

Option B (

\frac{1}{9}<I^2<\frac{1}{8}

) is correct because by analyzing the function within the integral and finding its bounds, the inequality for

I^2

is determined.

673 Easy

673 of 930

Analyzing +4 | -1

\lim\limits_{x \to 0} \frac{1}{x}\left\{ {\int \limits_y^a {e^{{{\sin }^2}t}}dt - \int \limits_{x + y}^a {e^{{{\sin }^2}t}}dt} \right\}

is equal to (where

a

is a constant)

View answer and explanation

Correct option: A

{e^{{{\sin }^2}y}}

Option A (

{e^{{{\sin }^2}y}}

) is correct because it is the result of applying the Leibniz rule for differentiating under the integral sign and then evaluating the limit.

674 Medium

674 of 930

Application +4 | -1

\lim\limits_{h \to 0} \frac{{{{\left( {a + h} \right)}^2}\sin \left( {a + h} \right) - {a^2}\sin a}}{h}

is equal to

View answer and explanation

Correct option: C

{a^2}\cos a + 2a\sin a

Option C (

{a^2}\cos a + 2a\sin a

) is correct because it is the result of applying the limit definition of the derivative to the given expression.

675 Easy

675 of 930

Application +4 | -1

\lim _{x \rightarrow+\infty} \frac{\sqrt{2 x^2+3}}{4 x+3}=

View answer and explanation

Correct option: B

\frac{1}{2 \sqrt{2}}

Option B (

\frac{1}{2 \sqrt{2}}

) is correct because after dividing by x and simplifying, the limit of the expression as x approaches infinity is

\frac{1}{2 \sqrt{2}}

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

676 Easy

676 of 930

Application +4 | -1

\lim _{x \rightarrow 3} \frac{\sqrt{3 x}-3}{\sqrt{2 x-4}-\sqrt{2}}

is equal to:

View answer and explanation

Correct option: B

\frac{1}{\sqrt{2}}

Option B (

\frac{1}{\sqrt{2}}

) is correct because it is the value of the limit after rationalizing and simplifying the expression.

677 Easy

677 of 930

Analyzing +4 | -1

Let

f(x)

be a polynomial function such that

f(x) + f'(x) + f''(x) = {x^5} + 64.

Then, the value of

\lim\limits_{x \to 1} \frac{{f(x)}}{{x - 1}}

is equal to

View answer and explanation

Correct option: A

– 15

Option B (– 60) is correct because by substituting a general 5th-degree polynomial and using L'Hopital's rule, the limit can be found to be -15.

678 Medium

678 of 930

Application +4 | -1

f(x)=\left(\frac{a+x}{b+x}\right)^{a+b+2 x}

, then

f^{\prime}(0)

is equal to

View answer and explanation

Correct option: B

\left(2 \log \frac{a}{b}+\frac{b^2-a^2}{a b}\right)\left(\frac{a}{b}\right)^{a+b}

Option B (

\left(2 \log \frac{a}{b}+\frac{b^2-a^2}{a b}\right)\left(\frac{a}{b}\right)^{a+b}

) is correct because it is the result of applying logarithmic differentiation and evaluating at x=0.

679 Easy

679 of 930

Application +4 | -1

The value of

\lim\limits_{x \to \infty } {a^x}\sin \left( {\frac{b}{{{a^x}}}} \right)

(a > 1)

View answer and explanation

Correct option: C

b

Option C (

b

) is correct because as

x

approaches infinity,

\frac{b}{{{a^x}}}

approaches 0, and the limit can be evaluated using the small angle approximation for sine.

680 Easy

680 of 930

Application +4 | -1

The value of

\lim \limits_{{x \rightarrow a}} \frac{x^m-a^m}{x^n-a^n}

, is

View answer and explanation

Correct option: A

\frac{m}{n} a^{m-n}

Option A (

\frac{m}{n} a^{m-n}

) is correct because applying L'Hopital's rule or using the standard limit formula yields the given result.

681 Easy

681 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 0}} \frac{1-\sqrt{\cos 2 x} \cdot \sqrt[3]{\cos 3 x} \cdot \sqrt[4]{\cos 4 x} \ldots \sqrt[n]{\cos n x}}{x^2}

has the value equal to 10 , the value of

n

View answer and explanation

Correct option: A

The correct answer is A (6) because by using L'Hopital's rule and simplifying the expression, the value of n can be determined by solving the resulting quadratic equation.

682 Medium

682 of 930

Analyzing +4 | -1

\lim\limits_{x \to 1} \frac{{\sin \left( {{e^x} - 1} \right)}}{{\log x}}

is equal to

View answer and explanation

Correct option: A

Option A (1) is correct because by using L'Hôpital's Rule and properties of limits, the given expression simplifies to 1.

683 Easy

683 of 930

Application +4 | -1

The value of

\lim \limits_{{x \rightarrow 1}}\left(\log _3 3 x\right)^{\log _x 3}

, is

View answer and explanation

Correct option: A

e

Option A (

e

) is correct because the limit can be simplified using logarithm properties and then evaluated using the exponential form of limits.

684 Medium

684 of 930

Analyzing +4 | -1

For each

x \in \mathbf{R}

, let

[x]

be greatest integer less than or equal to

x

. Then

\lim _{x \rightarrow 0} \frac{x([x]+|x|) \sin [x]}{x}

is equal to

View answer and explanation

Correct option: A

-\sin 1

Option A (

-\sin 1

) is correct because after evaluating the left-hand limit, the result is

-\sin 1

685 Easy

685 of 930

Application +4 | -1

\lim\limits_{x \to - \infty } \left( {3x + \sqrt {9{x^2} - x} } \right)

equals

View answer and explanation

Correct option: B

\frac{1}{6}

Option B (

\frac{1}{6}

) is correct because after rationalizing and simplifying, the limit evaluates to 1/6 as x approaches negative infinity.

686 Easy

686 of 930

Understanding +4 | -1

Let

f(x)= \begin{cases}\frac{\sin \left(x^2\right)}{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{cases}

Then

\lim _{x \rightarrow 0} f(x)

View answer and explanation

Correct option: B

Option B is correct because as x approaches 0, x² also approaches 0, and sin(x²)/x² approaches 1. Therefore, the limit becomes 0 * 1 = 0.

687 Easy

687 of 930

Application +4 | -1

f(x)=\left\{\begin{array}{ll}3 x-1, & x \geq 1 \\ 2 x+3, & x < 1\end{array}, g(x)=\left\{\begin{array}{cc}3-x, & x < 2 \\ 2 x-3, & x \geq 2\end{array}\right.\right.

, then

\lim \limits_{{x \rightarrow 2}} f(g(x))=

View answer and explanation

Correct option: B

Option B (2) is correct because the limit of

f(g(x))

x

approaches 2 is 2, based on the piecewise definitions of

f(x)

and

g(x)

688 Easy

688 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 0}} \frac{x^n \sin ^n x}{x^n-\sin ^n x}

is non-zero finite, then

n

is equal to

View answer and explanation

Correct option: B

Option B (2) is correct because after applying L'Hopital's rule multiple times, the limit becomes finite and non-zero only when n=2.

689 Medium

689 of 930

Analyzing +4 | -1

The value of

\lim\limits_{x \to \infty } {\left( {\frac{\pi }{2} - {{\tan }^{ - 1}}x} \right)^{1/x}}

View answer and explanation

Correct option: B

The correct answer is B (1) because the limit evaluates to 1 after applying L'Hopital's rule twice and simplifying the expression.

690 Easy

690 of 930

Analyzing +4 | -1

The value of

\lim\limits_{x \to 2} \frac{{x\left( {{5^x} - 1} \right)}}{{1 - \cos x}}

, is

View answer and explanation

Correct option: B

2\log 5

Option B (

2\log 5

) is correct because by applying L'Hopital's rule and limit properties, the expression simplifies to

2\log 5

Revision notes and key takeaways

Review LIMITS in your textbook or class notes, then use this set to check your understanding.

Showing 661-690 of 930 questions

Your Progress

Limits practice · Mixed difficulty

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice