The value of $\mathop {\lim }\limits_{x \to 3} \frac{{{3^x} - {x^2}}}{{{x^x} - {3^2}}}$, is

$\frac{{\log 3 - 1}}{{\log 3 + 1}}$

$\frac{{\log 3 + 1}}{{\log 3 - 1}}$

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JEE • MATHEMATICS • LIMITS

Limits practice · Easy

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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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Showing 421-450 of 647 questions

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421 Easy

421 of 647

Application +4 | -1

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

is equal to:

View answer and explanation

Correct option: C

e^{-5}

Option C (e^-5) is correct because the limit is of the form 1^infinity, which can be solved using the exponential limit formula.

422 Easy

422 of 647

Application +4 | -1

For

x \in R, \lim \limits_{{x \rightarrow \infty}}\left(\frac{x-3}{x+2}\right)^x=

View answer and explanation

Correct option: C

e^{-5}

Option C,

e^{-5}

, is correct because it is the result of rewriting the limit in the form of

e^{\lim x}

and then evaluating the limit.

423 Easy

423 of 647

Analyzing +4 | -1

f(x)= \begin{cases}\frac{\sin [x]}{[x]}, & {[x] \neq 0,} \\ 0, & \text [x] \neq 0,\end{cases}

where

[x]

denotes the greatest integer less than or equal to

x

. Then,

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

equals

View answer and explanation

Correct option: D

None of these

The correct answer is D (None of these) because the limit does not exist as the left-hand limit and right-hand limit are not equal.

424 Easy

424 of 647

Analyzing +4 | -1

\lim _{x \rightarrow 0} \frac{\sin ^2 x}{\sqrt{2}-\sqrt{1+\cos x}}

equals

View answer and explanation

Correct option: A

4 \sqrt{2}

Option A (

4 \sqrt{2}

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

4 \sqrt{2}

425 Easy

425 of 647

Analyzing +4 | -1

The value of

\lim\limits_{x \to 3} \frac{{{3^x} - {x^2}}}{{{x^x} - {3^2}}}

, is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the limit, after applying L'Hopital's rule, evaluates to a different expression than the given options.

426 Easy

426 of 647

Understanding +4 | -1

The value of

\lim \limits_{x \rightarrow 1^{-}}\left[\sin \sin ^{-1} x\right]

View answer and explanation

Correct option: A

Option A (0) is correct because as x approaches 1 from the left,

\sin(\sin^{-1}x)

approaches 1, and the greatest integer function of a value approaching 1 from below is 0.

427 Easy

427 of 647

Application +4 | -1

\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}-\frac{4}{x^2}\right)^{2 x}=e^3

, then 'a' is equal to :

View answer and explanation

Correct option: B

\frac{3}{2}

Option B (3/2) is correct because the limit can be evaluated using the exponential form of limits, leading to a = 3/2.

428 Easy

428 of 647

Application +4 | -1

f(x)=\lim \limits_{{n \rightarrow \infty}}\left[(1+x)\left(1+x^2\right)\left(1+x^4\right) \ldots n\right.

terms], then

f\left(\frac{1}{2}\right)=\ldots \ldots \quad|x| < 1

View answer and explanation

Correct option: B

Option B (2) is correct because by simplifying the infinite product using the difference of squares, we get a simple expression for f(x) and can evaluate f(1/2).

429 Easy

429 of 647

Analyzing +4 | -1

\lim _{x \rightarrow 1}\left(\frac{1}{1-x}-\frac{3}{1-x^3}\right)

is equal to

View answer and explanation

Correct option: D

-1

Option D (-1) is correct because after combining the fractions and factoring, the limit can be evaluated using L'Hopital's rule or by direct substitution.

430 Easy

430 of 647

Application +4 | -1

\lim \limits_{{x \rightarrow 2}} \frac{x^{n}-2^{n}}{x-2}=80

, where

n

is a positive integer, then

n=

View answer and explanation

Correct option: B

Option B (5) is correct because solving the limit equation

\lim \limits_{{x \rightarrow 2}} \frac{x^{n}-2^{n}}{x-2}=80

yields

n=5

431 Easy

431 of 647

Analyzing +4 | -1

The value of

\lim\limits_{x \to 0} \frac{{\sqrt[3]{{1 + \sin x}} - \sqrt[3]{{1 - \sin x}}}}{x}

, is

View answer and explanation

Correct option: A

\frac{2}{3}

Option A (

\frac{2}{3}

) is correct because the limit is evaluated using series expansion or L'Hopital's rule.

432 Easy

432 of 647

Application +4 | -1

If value of

\lim \limits_{{x \rightarrow 0}} \frac{\sqrt{2+x}-\sqrt{2}}{x}

is equal to

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

then '

a

' equals

View answer and explanation

Correct option: A

Option A is correct because the limit evaluates to 1/(2√2), implying that 'a' equals 2.

433 Easy

433 of 647

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 0}} \frac{x \sin (\sin x)-\sin ^2 x}{x^6}

equals

View answer and explanation

Correct option: C

\frac{1}{18}

The correct answer is C (

\frac{1}{18}

) because after expanding the trigonometric functions using their Taylor series and simplifying, the limit equals

\frac{1}{18}

434 Easy

434 of 647

Understanding +4 | -1

f(x)=x^2 \sin \frac{1}{x}

, where

x \neq 0

, then the value of the function

{f}

{x}=0

, so that the function is continuous at

x=0

, is

View answer and explanation

Correct option: A

Option A is correct because the limit of the function as x approaches 0 is 0, making the function continuous at x=0 if f(0)=0.

435 Easy

435 of 647

Analyzing +4 | -1

\lim \limits_{x \rightarrow 1^{+}} \frac{\sqrt{x^2-1}+\sqrt{x-1}}{\sqrt{x^2-1}}

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because after simplifying the limit expression by factoring and cancelling, the limit can be directly evaluated.

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436 Easy

436 of 647

Application +4 | -1

Let

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

. Then,

\log _e P

, is equal to

View answer and explanation

Correct option: B

\frac{1}{2}

Option B (

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

) is correct because the limit can be evaluated using the exponential form of limits and the standard limit

\lim_{x \rightarrow 0} \frac{{\tan x}}{{x}} = 1

437 Easy

437 of 647

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 0}}\left(\frac{2}{x^3}\left(\sin ^{-1} x-\tan ^{-1} x\right)\right)^{2 / x^2}

equals

View answer and explanation

Correct option: D

\frac{1}{\sqrt{e}}

The correct answer is D (

\frac{{1}}{{\sqrt{{e}}}}

) because by using Taylor series expansions for

\sin^{-1}x

and

\tan^{-1}x

, the limit can be evaluated to

e^{-1/2}

438 Easy

438 of 647

Analyzing +4 | -1

\lim _{x \rightarrow 0} \frac{12^x-3^x-4^x+1}{\sqrt{2 \cos x+7}-3}=

View answer and explanation

Correct option: D

-6 \log _e 3 \cdot \log _e 4

Option D (

-6 \log _e 3 \cdot \log _e 4

) is correct because the limit can be evaluated by factoring the numerator and using standard limit formulas.

439 Easy

439 of 647

Application +4 | -1

\lim\limits_{\theta \to \frac{\pi }{2}} \,\,\frac{{\frac{\pi }{2} - \theta }}{{\cot \theta }}

is equal to

View answer and explanation

Correct option: C

Option C (1) is correct because by applying L'Hopital's rule or converting cot to sin/cos, the limit evaluates to 1.

440 Easy

440 of 647

Analyzing +4 | -1

The value of

\lim\limits_{x \to 2} \frac{5}{{\sqrt 2 - \sqrt x }}

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, indicating the limit does not exist.

441 Easy

441 of 647

Application +4 | -1

The value of

\lim\limits_{x \to 0} \frac{{{e^{ax}} - {e^{bx}}}}{x}

is equal to

View answer and explanation

Correct option: B

a - b

Option B (

a - b

) is correct because it is the result of applying L'Hopital's rule or using series expansion to evaluate the limit.

442 Easy

442 of 647

Application +4 | -1

f:R \to R

is defined by

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\frac{{x - 2}}{{{x^2} - 3x + 2}}}&{,\mathrm{if}}&{x \in R - \left\{ {1,2} \right\}}\\ 2&{,\mathrm{if}}&{x = 1}\\ 1&{,\,\mathrm{if}}&{x = 2} \end{array}} \right.

Then

\lim\limits_{x \to 2} \frac{{f\left( x \right) - f\left( 2 \right)}}{{x - 2}}

is equal to

View answer and explanation

Correct option: B

- 1

Option B (-1) is correct because the limit of the given expression, after simplification and factorization, evaluates to -1.

443 Easy

443 of 647

Application +4 | -1

\lim\limits_{x \to 1} \frac{{{x^m} - 1}}{{{x^n} - 1}}

is equal to

View answer and explanation

Correct option: B

\frac{m}{n}

Option B (

\frac{m}{n}

) is correct because by applying L'Hopital's rule, the limit evaluates to m/n.

444 Easy

444 of 647

Application +4 | -1

The value of

\lim \limits_{{x \rightarrow 0}} \frac{\int\limits_{0}^x \cos t d t}{x}

View answer and explanation

Correct option: D

None of these

The correct answer is None of these, as the limit evaluates to 1 using L'Hopital's rule.

445 Easy

445 of 647

Application +4 | -1

\lim\limits _{x \rightarrow 2^{+}}\left\{\frac{[x]^3}{3}-\left[\frac{x}{3}\right]^3\right\}

is equal to

View answer and explanation

Correct option: C

\frac{8}{3}

Option C (

\frac{8}{3}

) is correct because it is the result of evaluating the limit as x approaches 2 from the right, considering the greatest integer function.

446 Easy

446 of 647

Application +4 | -1

\lim \limits_{{x \rightarrow \infty}}\left(\frac{x^2+5 x+3}{x^2+x+2}\right)^x

View answer and explanation

Correct option: A

e^4

Option A (

e^4

) is correct because the limit is evaluated using the standard limit formula and algebraic manipulation.

447 Easy

447 of 647

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

448 Easy

448 of 647

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.

449 Easy

449 of 647

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}

450 Easy

450 of 647

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.

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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