$\lim _\limits{x \rightarrow 0}\left(x^n \log _e x\right)$ is

does not exists either finitely or infinitely

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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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Showing 181-210 of 930 questions

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181 Easy

181 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow \pi / 2}} \frac{\left[\frac{x}{2}\right]}{\log _e(\sin x)}

(where [.] denotes the greatest integer function)

View answer and explanation

Correct option: C

Equals 0

Option C (Equals 0) is correct because both the left-hand limit and the right-hand limit of the given expression as x approaches

\pi/2

are equal to 0.

182 Easy

182 of 930

Analyzing +4 | -1

Let

f:R \to R

be a differentiable function satisfying

f'(3) + f'(2) = 0.

then

\lim\limits_{x \to 0} {\left( {\frac{{1 + f(3 + x) - f(3)}}{{1 + f(2 - x) - f(2)}}} \right)^{\frac{1}{x}}}

is equal to

View answer and explanation

Correct option: D

Option D (1) is correct because applying L'Hopital's rule and using the given condition

f'(3) + f'(2) = 0

simplifies the limit to 1.

183 Easy

183 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 2 a}} \frac{\sqrt{x-2 a}+\sqrt{x}-\sqrt{2 a}}{\sqrt{x^2-4 a^2}}

, is

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the limit is evaluated by rationalizing the numerator and simplifying the expression.

184 Easy

184 of 930

Application +4 | -1

\lim \limits_{{x \rightarrow 0}} \frac{\sqrt{1+x}-1}{x}

is equal to

View answer and explanation

Correct option: A

1 / 2

Option A (1/2) is correct because rationalizing the numerator and simplifying the expression leads to a limit of 1/2 as x approaches 0.

185 Easy

185 of 930

Application +4 | -1

Let

f(x)=4

and

f^{\prime}(x)=4

. Then

\lim _{x \rightarrow 2} \frac{x f(2)-2 f(x)}{x-2}

is given by

View answer and explanation

Correct option: C

-4

Option C (-4) is correct because applying L'Hopital's rule gives the correct limit value.

186 Medium

186 of 930

Application +4 | -1

The value of

\lim\limits_{x \to {0^ + }} {x^m}{\left( {\log x} \right)^n},m,n.\mathrm{ }N

View answer and explanation

Correct option: A

Option A (0) is correct because by applying L'Hopital's rule repeatedly, the limit evaluates to 0 as x approaches 0 from the positive side.

187 Easy

187 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow e}} \frac{\log _e x-1}{|x-e|}

, is

View answer and explanation

Correct option: D

Non-existent

Option D (Non-existent) is correct because the left-hand limit and the right-hand limit are not equal, indicating that the limit does not exist.

188 Easy

188 of 930

Application +4 | -1

The value of

\lim \limits_{{x \rightarrow a}}\left(2-\frac{a}{x}\right)^{\tan \frac{\pi x}{2 a}}

, is

View answer and explanation

Correct option: C

e^{-2 / \pi}

Option C (

e^{-2 / \pi}

) is correct because the limit can be evaluated using the exponential form of limits and L'Hopital's rule after substitution.

189 Easy

189 of 930

Application +4 | -1

Value of

\lim \limits_{{x \rightarrow 0}} \frac{\ell n \left(1+2 x-3 x^2+4 x^3\right)}{\ell n \left(1-5 x+6 x^2-7 x^3\right)}

is -

View answer and explanation

Correct option: C

-\frac{2}{5}

Option C (-2/5) is correct because the limit is evaluated using L'Hopital's rule or by manipulating the logarithmic expressions.

190 Medium

190 of 930

Analyzing +4 | -1

f(x)=\left\{\begin{array}{cl}\frac{[x]^2+\sin [x]}{[x]} & \text { for }[x] \neq 0 \\ 0 & \text { for }[x]=0\end{array}\right.

where

[x]

denotes the greatest integer function, then

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

, is

View answer and explanation

Correct option: D

Non-existent

Option D (Non-existent) is correct because the left-hand limit and right-hand limit are not equal, so the limit does not exist.

191 Medium

191 of 930

Analyzing +4 | -1

Let

f(x+y)=f(x)+f(y)

and

f(x)=x^2 g(x)

for all

x, y \in R

, where

g(x)

is continuous function. Then

f^{\prime}(x)

is equal to

View answer and explanation

Correct option: D

Option D is correct because using the given functional equation and the definition of the derivative, we find that f'(x) = 0.

192 Hard

192 of 930

Application +4 | -1

Let

f:R \to R

be a differentiable function such that

f\left( 2 \right) = 2

. Then, the value of

\lim\limits_{x \to 2} \int \limits_2^{f\left( x \right)} \frac{{4\;{t^3}}}{{x - 2}}dt

, is

View answer and explanation

Correct option: C

32f'\left( 2 \right)

Option C (

32f'(2)

) is correct because it is the result of applying L'Hopital's rule and the fundamental theorem of calculus to the given limit.

193 Medium

193 of 930

Application +4 | -1

Which of the following is/are true?

I. The derivative of

{f}({x})={x}^3

{x}^2

II. The derivative of

f(x)=\frac{1}{x^3}

\frac{-1}{x^2}

View answer and explanation

Correct option: D

Both I and II are false

Option D is correct because both statements are false. The derivative of x³ is 3x², and the derivative of 1/x³ is -3/x⁴.

194 Medium

194 of 930

Analyzing +4 | -1

\underset{x \rightarrow a-}{L t}\left(\frac{|x|^3}{a}-\left[\frac{x}{a}\right]^3\right)(a>0)

when

[x]

is G.I.F. is

View answer and explanation

Correct option: D

Option D (0) is correct because as x approaches a from the left, the greatest integer function evaluates to 0, making the entire expression equal to 0.

195 Medium

195 of 930

Analyzing +4 | -1

The values of

a

and

b

such that

\lim\limits_{x \to 0} \frac{{x\left( {1 + a\cos x} \right) - b\sin x}}{{{x^3}}} = 0

, are

View answer and explanation

Correct option: C

- \frac{5}{2}, - \frac{3}{2}

Option C (

\frac{{ - 5}}{2}, - \frac{3}{2}

) is correct because by expanding the trigonometric functions using Taylor series and equating coefficients, the values of a and b can be determined.

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196 Easy

196 of 930

Understanding +4 | -1

The value of

\lim \limits_{x \rightarrow 0^{-}}\left[\frac{\sin x}{x}\right]

View answer and explanation

Correct option: B

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

\frac{{\sin x}}{{x}}

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

197 Easy

197 of 930

Application +4 | -1

The derivative of function

f\left( x \right)

{\tan ^4}x

. If

f\left( x \right) = 0

, then

\lim\limits_{x \to 0} \frac{{f\left( x \right)}}{x}

is equal to

View answer and explanation

Correct option: B

Option B (0) is correct because by applying L'Hopital's rule, the limit is equal to the limit of the derivative of the function, which is 0.

198 Medium

198 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 0}} \frac{\int\limits_{0}^{x^2} \cos t^2 d t}{x \sin x}

View answer and explanation

Correct option: B

Option B (1) is correct because after applying L'Hopital's rule twice, the limit evaluates to 1.

199 Medium

199 of 930

Application +4 | -1

The value of

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

, is

View answer and explanation

Correct option: C

\frac{2}{\pi }

Option C (

\frac{2}{\pi }

) is correct because the limit is evaluated using L'Hopital's rule or by rewriting the expression using trigonometric identities and then applying standard limits.

200 Easy

200 of 930

Application +4 | -1

The value of

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

, is

View answer and explanation

Correct option: C

{e^{ - 2}}

Option C (

{e^{-2}}

) is correct because it is the result of manipulating the expression to fit the standard limit definition of e and then evaluating the limit.

201 Easy

201 of 930

Understanding +4 | -1

The value of

\lim\limits_{x \to 0} {\left( {\frac{{x + 3}}{{x + 1}}} \right)^x}

View answer and explanation

Correct option: C

Option C (1) is correct because as x approaches 0, the expression (

\frac{{x + 3}}{{x + 1}}

)

^x

approaches (

\frac{{3}}{{1}}

)

^0

, which equals 1.

202 Medium

202 of 930

Application +4 | -1

The value of

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

View answer and explanation

Correct option: C

\frac{3}{2}

Option D (

\frac{3}{2}

) is correct because by factoring and using trigonometric limit properties, the limit evaluates to 3/2.

203 Medium

203 of 930

Analyzing +4 | -1

Let

P_n=\frac{2^3-1}{2^3+1} \cdot \frac{3^3-1}{3^3+1}+\cdots+\frac{n^3-1}{n^3+1} ; n=2,3,4, \ldots

Then

\lim \limits_{{n \rightarrow \infty}} P_n

is equal to

View answer and explanation

Correct option: D

\frac{2}{3}

Option D (

\frac{2}{3}

) is correct because after factoring and telescoping the product, the limit as n approaches infinity is

\frac{2}{3}

204 Easy

204 of 930

Application +4 | -1

Let

f\left( x \right) = x\left[ x \right]

, where [x] denotes integral part of

x

. If

a

is not an integer, then

\lim\limits_{h \to 0} \frac{{f\left( {a + h} \right) - f\left( a \right)}}{h} =

View answer and explanation

Correct option: C

[a]

Option C ([a]) is correct because since a is not an integer, for small h, [a+h] = [a], simplifying the limit calculation.

205 Easy

205 of 930

Understanding +4 | -1

\lim \limits_{{x \rightarrow 1}} \frac{1}{|1-x|}=

View answer and explanation

Correct option: D

\infty

Option D (

\infty

) is correct because as x approaches 1, the absolute value of (1-x) approaches 0, and the reciprocal approaches infinity.

206 Easy

206 of 930

Analyzing +4 | -1

Let

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\left( {\frac{1}{2}} \right)\left\{ {g\left( x \right) + \left( x \right)} \right\}\sin \left( x \right),x \ge 1}\\ {\sin x/x,\,\,\,x < 1} \end{array}} \right.

Where

g\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {1,\mathrm{if}\,\,x > 0}\\ { - 1,\mathrm{if}\,\,x < 0}\\ {0,\mathrm{if}\,\,x = 0} \end{array}} \right.

Then,

\lim\limits_{x \to 1} f\left( x \right)

is equal to

View answer and explanation

Correct option: C

\sin 1

Option C (sin 1) is correct because both the left-hand limit and right-hand limit of the function as x approaches 1 are equal to sin 1.

207 Medium

207 of 930

Application +4 | -1

\lim \limits_{{x \rightarrow 0}}\left(x^n \log _e x\right)

View answer and explanation

Correct option: C

The correct answer is C (0) because rewriting the expression and applying L'Hôpital's rule yields the value 0.

208 Easy

208 of 930

Application +4 | -1

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

equals

View answer and explanation

Correct option: C

e^2

Option C (

e^2

) is correct because the limit can be evaluated using the standard limit

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

209 Easy

209 of 930

Analyzing +4 | -1

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

View answer and explanation

Correct option: B

\frac{1}{2}

Option B (

\frac{1}{2}

) is correct because after applying trigonometric identities and limit properties, the expression simplifies to 1/2.

210 Easy

210 of 930

Analyzing +4 | -1

The value of

\lim\limits_{x \to - \pi } \frac{{\left| {x + \pi } \right|}}{{\sin x}}

View answer and explanation

Correct option: D

Does not exist

Option D (Does not exist) is correct because the left-hand limit and the right-hand limit at x = -π are not equal, indicating that the limit does not exist.

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