If $f\left( x \right) = \mathop {\lim }\limits_{y \to x} \,\,\frac{{{{\sin }^2}y\,\, - \,\,{{\sin }^2}\,\,x}}{{{y^2}\,\, - \,\,{x^2}}}$, then $\smallint 4x\,\,\;f\left( x \right)\,\,dx$ is equal to

$ - \,2\,\,\cos \,\,2x\,\, + \,\,c$

If $\lim _{x \rightarrow 0}\left(1+a x+b x^2\right)^{1 / x}=e^3$, then

$a=3 / 2$ and $b$ is any real number

$b=3$ and $a$ is any real number

$a=1$ and $b$ is any real number

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JEE • MATHEMATICS • LIMITS

Limits practice · Mixed difficulty

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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.

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571 Easy

571 of 930

Application +4 | -1

f\left( a \right) = 2,f'\left( a \right) = 1,g\left( a \right) = 3,g'\left( a \right) = - 1

, then

\lim\limits_{x \to a} \frac{{f\left( a \right)g\left( x \right) - f\left( x \right)g\left( a \right)}}{{x - a}}

is equal to

View answer and explanation

Correct option: D

- 5

Option D (-5) is correct because applying L'Hopital's rule and substituting the given values results in -5.

572 Easy

572 of 930

Application +4 | -1

The value of

\lim \limits_{\substack{x \rightarrow 1 \\ y \rightarrow 0}} \frac{y^3}{x^3-y^2-1}

(x, y) \rightarrow(1,0)

along the line

y=x-1

is-

View answer and explanation

Correct option: A

Option A (0) is correct because substituting y = x - 1 and taking the limit as y approaches 0 results in a value of 0.

573 Medium

573 of 930

Application +4 | -1

A non-zero function

f

satisfies

f(x-y)=\frac{f(x)}{f(y)}, \forall x, y \in \mathbf{R}

. If

f(0)=5

and

f(5)=5

, then

f(5)=\ldots \ldots

View answer and explanation

Correct option: A

Option A (1) is correct because by substituting x=5 and y=0 into the given functional equation, we can derive that f(5) = 1.

574 Medium

574 of 930

Understanding +4 | -1

Assertion:

\lim \limits_{{x \rightarrow 0}}(1+3 x)^{1 / x}=e^3

.

Reason: Since

\lim \limits_{{x \rightarrow 0}}(1+x)^{1 / x}=e

View answer and explanation

Correct option: A

Assertion is correct, reason is correct; reason is a correct
explanation for assertion

Option A (Assertion is correct, reason is correct; reason is a correct explanation for assertion) is correct because the limit can be evaluated using the given standard limit.

575 Easy

575 of 930

Analyzing +4 | -1

f\left( x \right) = \lim\limits_{y \to x} \,\,\frac{{{{\sin }^2}y\,\, - \,\,{{\sin }^2}\,\,x}}{{{y^2}\,\, - \,\,{x^2}}}

, then

\int 4x\,\,\;f\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: C

- \,\cos \,\,2x\,\, + \,\,c

Option C is correct because after applying L'Hôpital's rule and integrating, the result is

- \,\cos \,2x\,\, + \,\,c

576 Medium

576 of 930

Analyzing +4 | -1

In a circle of radius

r

, an isosceles

\Delta A B C

is inscribed with

A B=A C

. If

\Delta A B C

has perimeter

p=2\left[\sqrt{2 h r-h^2}+\sqrt{2 h r}\right]

and area

A=h \sqrt{2 h r-h^2}

, where

h

is the altitude from

A

B C

, then

\lim \limits_{h \rightarrow 0^{+}} \frac{A}{p^3}

View answer and explanation

Correct option: B

\frac{1}{128 r}

The correct answer is B (1/128r) because by simplifying the expression and evaluating the limit as h approaches 0, the result is 1/(128r).

577 Easy

577 of 930

Analyzing +4 | -1

Let

f\left( a \right) = g\left( a \right) = k

and their nth derivatives

{f^n}\left( a \right),{g^n}\left( a \right)

exist and are not equal for somen. Further if

\lim\limits_{x \to a} \frac{{f\left( a \right)g\left( x \right) - f\left( a \right) - g\left( a \right)f\left( x \right) + g\left( a \right)}}{{g\left( x \right) - f\left( x \right)}} = 4

, then the value of

k

is equal to

View answer and explanation

Correct option: A

Option A (4) is correct because by applying L'Hôpital's rule, the limit simplifies to k, which is equal to 4.

578 Easy

578 of 930

Application +4 | -1

\lim\limits_{x \to 0} \frac{{2{{\sin }^2}3x}}{{{x^2}}}

is equal to

View answer and explanation

Correct option: C

Option C (18) is correct because by using the limit

\lim_{x \to 0} \frac{{\sin x}}{x} = 1

, the given limit simplifies to 18.

579 Easy

579 of 930

Application +4 | -1

Let

f(x)=\left\{\begin{array}{cc}\frac{\sin 4 x}{\log (1+3 x)}, & x \neq 0 \\ A+1, & x=0\end{array}\right.

The value of

A

for

f

to be continuous at

x=0

View answer and explanation

Correct option: B

\frac{1}{3}

Option B (

\frac{1}{3}

) is correct because for

f

to be continuous at

x=0

A+1 = \lim_{x \rightarrow 0} \frac{\sin 4x}{\log(1+3x)} = \frac{4}{3}

, which gives

A = \frac{1}{3}

580 Easy

580 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 0}} \frac{2 \sin x-\sin 2 x}{\tan ^3 x}

is equal to

View answer and explanation

Correct option: B

Option B (1) is correct because after simplifying the expression using trigonometric identities, the limit evaluates to 1.

581 Medium

581 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 0}}\left(\frac{\pi}{4 x}-\frac{\pi}{2 x\left(e^{\pi x}+1\right)}\right)

equals

View answer and explanation

Correct option: D

\frac{\pi^2}{8}

Option D (

\frac{\pi^2}{8}

) is correct because after simplifying the expression and applying L'Hôpital's rule, the limit evaluates to

\frac{\pi^2}{8}

582 Easy

582 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 1}}\left\{\frac{x^n-1}{n(x-1)}\right\}^{\frac{1}{x-1}}

, is

View answer and explanation

Correct option: C

e^{\frac{n-1}{2}}

Option C (

e^{\frac{n-1}{2}}

) is correct because it is obtained by simplifying the expression using algebraic manipulation and then applying limit properties.

583 Easy

583 of 930

Application +4 | -1

\lim _{x \rightarrow 1} \frac{x^4-1}{x-1}=\lim _{x \rightarrow k} \frac{x^3-k^3}{x^2-k^2}

, then

k

is:

View answer and explanation

Correct option: A

\frac{8}{3}

Option A (

\frac{8}{3}

) is correct because it is the value of k that satisfies the given limit equation after applying L'Hopital's Rule.

584 Medium

584 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow \infty}} x\left[\tan ^{-1}\left(\frac{x+1}{x+2}\right)-\tan ^{-1}\left(\frac{x}{x+2}\right)\right]

View answer and explanation

Correct option: C

\frac{1}{2}

Option C (

\frac{1}{2}

) is correct because by using the arctangent subtraction formula and the limit tan^-1(x)/x as x approaches 0, the limit evaluates to

\frac{1}{2}

585 Easy

585 of 930

Application +4 | -1

\lim \limits_{x \rightarrow \sqrt{3}} \frac{x^2-3}{x^2+3 \sqrt{3} x-12}

is equal to

View answer and explanation

Correct option: B

\frac{2}{5}

Option B (

\frac{2}{5}

) is correct because it is the result of factoring and simplifying the limit expression as x approaches

\sqrt{3}

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586 Medium

586 of 930

Application +4 | -1

\lim _{x \rightarrow 0}\left(1+a x+b x^2\right)^{1 / x}=e^3

, then

View answer and explanation

Correct option: A

a=3

and

b

is any real number

Option A (

a=3

and

b

is any real number) is the correct answer because the limit simplifies to

e^a

, which must equal

e^3

, therefore

a=3

587 Easy

587 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow-1}}\left(\frac{x^4+x^2+x+1}{x^2-x+1}\right)^{\frac{1-\cos (x+1)}{(x+1)^2}}

is equal to

View answer and explanation

Correct option: B

\sqrt{2 / 3}

Option B (

\sqrt{2 / 3}

) is correct because by substituting

x = -1

into the base and using the limit

\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}

for the exponent, the limit evaluates to

\sqrt{2/3}

588 Easy

588 of 930

Analyzing +4 | -1

\lim\limits_{n \to \infty } \frac{{{{3.2}^{n + 1}} - {{4.5}^{n + 1}}}}{{{{5.2}^n} + {{7.5}^n}}}

is equal to

View answer and explanation

Correct option: C

- \frac{{20}}{7}

The correct answer is C (

\frac{{ - 20}}{7}

) because the limit is evaluated by dividing by the dominant term and simplifying the expression.

589 Easy

589 of 930

Analyzing +4 | -1

\lim \limits_ {x \rightarrow 0^{-}} \frac{[x]+\left[x^2\right]+\left[x^3\right]+\ldots .+\left[x^{2 n+1}\right]+n+1}{1+\left[x^2\right]+|x|+2 x}, n \in N

is equal to

View answer and explanation

Correct option: D

Option D (0) is correct because as x approaches 0 from the left, the numerator becomes 0 and the denominator approaches 1, resulting in a limit of 0.

590 Medium

590 of 930

Application +4 | -1

Let

p=\lim _{x \rightarrow 0^{+}}\left(1+\tan ^2 \sqrt{x}\right)^{\frac{1}{2 x}}

then

\log p

is equal to :

View answer and explanation

Correct option: A

\frac{1}{2}

Option A (

\frac{1}{2}

) is correct because after taking the logarithm and applying L'Hopital's rule, the limit evaluates to

\frac{1}{2}

591 Medium

591 of 930

Application +4 | -1

\lim _{x \rightarrow+\infty} \frac{\sqrt{9 x^2+2}-x}{4 x+11}=

View answer and explanation

Correct option: D

1 / 2

Option D (

\frac{1}{2}

) is correct because, after dividing by x and simplifying, the limit of the expression as x approaches infinity is 1/2.

592 Medium

592 of 930

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\frac{2}{\pi }

Option C (

\frac{2}{\pi }

) is correct because after applying L'Hopital's rule and trigonometric identities, the limit evaluates to

\frac{2}{\pi }

593 Easy

593 of 930

Analyzing +4 | -1

\lim\limits_{x \to 0} \frac{{{{\sin }^{ - 1}}x - x}}{{{x^3}\cos x}}

is equal to

View answer and explanation

Correct option: C

\frac{1}{6}

Option D (None of these) is correct because the limit does not exist since the function oscillates between 0 and 1 as x approaches 0.

594 Easy

594 of 930

Analyzing +4 | -1

\lim \limits_{{t \rightarrow 0}} \frac{1-(1+t)^t}{\ln (1+t)-t}

is equal to

View answer and explanation

Correct option: C

Option C (2) is correct because after applying L'Hopital's rule or using series expansion, the limit evaluates to 2.

595 Easy

595 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 1}}\left[\tan \left(\frac{\pi}{4}+\log x\right)\right]^{\frac{1}{\log x}}

View answer and explanation

Correct option: D

\mathrm{e}^2

Option D (

e^2

) is correct because by applying L'Hôpital's Rule after transforming the limit into an exponential form, the limit evaluates to

e^2

596 Easy

596 of 930

Understanding +4 | -1

f(x)=\left\{\begin{array}{cc}2 x+3, & x \leq 0 \\ 3(x+1), & x>0\end{array}\right.

then the value of

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

View answer and explanation

Correct option: A

Option A (3) is correct because both the left-hand limit and the right-hand limit as x approaches 0 are equal to 3.

597 Easy

597 of 930

Analyzing +4 | -1

\lim _{x \rightarrow 0} \frac{a^{\tan x}-a^{\sin x}}{\tan x-\sin x}

is equal to

(a>0)

View answer and explanation

Correct option: A

\log _e a

Option A (

\log _e a

) is correct because the limit can be evaluated by recognizing it as the derivative of

a^x

and applying L'Hopital's rule or a substitution.

598 Easy

598 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 1}} \frac{a x^2+b x+c}{(x-1)^2}=2

, then

\lim \limits_{{x \rightarrow 1}} \frac{(x-a)(x-b)(x-c)}{x+1}

, is

View answer and explanation

Correct option: B

\frac{5}{2}

Option B (

\frac{5}{2}

) is correct because by using L'Hopital's rule and the given limit, we can find the values of a, b, and c and then evaluate the required limit.

599 Easy

599 of 930

Analyzing +4 | -1

f(x)=x^m, m

being a non-negative integer, then the value of

m

for which

f^{\prime}(\alpha+\beta)=f^{\prime}(\alpha)+f^{\prime}(\beta)

, for all

\alpha, \beta > 0

, is

View answer and explanation

Correct option: B

Option B (2) is correct. By substituting the given condition into the derivative of f(x) and comparing terms, we find that m = 2 satisfies the equation.

600 Hard

600 of 930

Analyzing +4 | -1

\lim\limits_{x \to 3} \,\left( {\left[ {x - 3} \right] + \left[ {3 - x} \right] - x} \right),

where

\left[ \cdot \right]

denotes the greatest integer function, is equal to

View answer and explanation

Correct option: B

-4

Option B (-4) is correct because the greatest integer function simplifies to -1, and the limit can be directly evaluated.

Revision notes and key takeaways

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

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