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 · Medium

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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 267 questions

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181 Medium

181 of 267

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}

182 Medium

182 of 267

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

183 Medium

183 of 267

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}

184 Medium

184 of 267

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.

185 Medium

185 of 267

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 }

186 Medium

186 of 267

Analyzing +4 | -1

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

, then

a,\;b

are

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the given condition leads to a = b, and none of the provided options satisfy this condition.

187 Medium

187 of 267

Analyzing +4 | -1

l = \lim\limits_{x \to - 2} \frac{{\tan \pi \mathrm{ }x}}{{x + 2}} + \lim\limits_{x \to \infty } {\left( {1 + \frac{1}{{{x^2}}}} \right)^x}

then which one of the following is not correct?

View answer and explanation

Correct option: C

l < 4

Option

l < 4

is not correct because the value of

l

\pi + 1

, which is greater than 4.

188 Medium

188 of 267

Application +4 | -1

\lim \limits_{{n \rightarrow \infty}} n \cos \left(\frac{\pi}{4 n}\right) \sin \left(\frac{\pi}{4 n}\right)=k

, then

k

View answer and explanation

Correct option: A

\frac{\pi}{4}

Option A (

\frac{{\pi}}{{4}}

) is correct because applying the limit and using the small angle approximation yields

\frac{{\pi}}{{4}}

189 Medium

189 of 267

Analyzing +4 | -1

\lim\limits_{x \to \infty } \left( {\sqrt {x + \sqrt {x + \sqrt x } } - \sqrt x } \right)

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}

Option A (1/2) is correct because after rationalizing and simplifying, the limit as x approaches infinity evaluates to 1/2.

190 Medium

190 of 267

Analyzing +4 | -1

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

View answer and explanation

Correct option: B

{e^{2/\pi }}

Option B (

{e^{2/\pi }}

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

191 Medium

191 of 267

Application +4 | -1

\lim \limits_{{x \rightarrow \infty}}\left(\frac{x+7}{x+2}\right)^{x+4}

View answer and explanation

Correct option: D

e^5

Option D (

e^5

) is correct because it's the result of evaluating the limit using the standard limit formula for exponential functions.

192 Medium

192 of 267

Analyzing +4 | -1

The value of

\lim\limits_{n \to \infty } {\left( {\cos \frac{x}{n}} \right)^n}

, is

View answer and explanation

Correct option: C

Option C (1) is correct because the limit can be evaluated by rewriting the expression using the exponential function and L'Hopital's rule or series expansion.

193 Medium

193 of 267

Application +4 | -1

The value of

\lim \limits_{{h \rightarrow 0}} \frac{f(x+h)-f(x)}{h}

, where

f(x)=\sin (x+1)

, is

View answer and explanation

Correct option: B

\cos (x+1)

Option B (

\cos (x+1)

) is correct because it is the derivative of sin(x+1) with respect to x.

194 Medium

194 of 267

Analyzing +4 | -1

\lim _{x \rightarrow 0}\left(\tan \left(\frac{\pi}{4}+x\right)\right)^{1 / x}

is equal to:

View answer and explanation

Correct option: D

e^2

Option D (

e^2

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

e^{\lim_{x \to 0} f(x)}

and the exponent evaluates to 2.

195 Medium

195 of 267

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 0}}\left(\left[\frac{11 x}{\sin x}\right]+\left[\frac{21 \sin x}{x}\right]\right)

, where

[x]

is the greatest integer less than or equal to

x

View answer and explanation

Correct option: B

Option B (31) is correct because as x approaches 0, the greatest integer functions evaluate to 11 and 20 respectively, summing to 31.

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196 Medium

196 of 267

Analyzing +4 | -1

The value of

\lim\limits_{x \to 0} \,\,\frac{{\int_t^x {t\mathrm{ }dt} \,\,}}{{x\tan \left( {x + \pi } \right)}}

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}

Option C (

\frac{1}{2}

) is correct because, by applying L'Hopital's rule or direct integration and simplification, the limit evaluates to 1/2.

197 Medium

197 of 267

Analyzing +4 | -1

Suppose

x

is real. Define

f(x)

\frac{x}{2}+1

\begin{aligned} & f^{(2)}(x)=f(f(x)) \\ & f^{(3)}(x)=f\left(f^{(2)}(x)\right) \end{aligned}

and in general

f^{n+1}(x)=f\left(f^{(n)}(x)\right)

for

n \geq 1

where

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

Then

\lim \limits_{{n \rightarrow-\infty}} f^{(n)}(x)

is equal to

View answer and explanation

Correct option: B

Option B (2) is correct because the limit of the recursively defined function

f^{(n)}(x)

as n approaches negative infinity converges to 2.

198 Medium

198 of 267

Analyzing +4 | -1

Let

x_0=0, x_1=1

and

x_{n+1}=x_n+\sqrt{1+x_n^2}

for

n \geq 1

.
Then

\lim \limits_{{n \rightarrow \infty}}\left(\frac{x_n}{2^{n-1}}\right)=

View answer and explanation

Correct option: D

\frac{4}{\pi}

Option D is correct. By using the recurrence relation and trigonometric identities, the limit can be evaluated to 4/π.

199 Medium

199 of 267

Application +4 | -1

f: R \rightarrow R

is given by

f(x)=x+1

, then the value of

\lim \limits_{{n \rightarrow \infty}} \frac{1}{n}\left[f(0)+f\left(\frac{5}{n}\right)+f\left(\frac{10}{n}\right)+\ldots+f\left(\frac{5(n-1)}{n}\right)\right], \text { is : }

View answer and explanation

Correct option: D

\frac{7}{2}

Option D (

\frac{{7}}{{2}}

) is correct because the limit represents a Riemann sum which can be evaluated as a definite integral.

200 Medium

200 of 267

Analyzing +4 | -1

\lim \limits_{{n \rightarrow \infty}}\left(1+\frac{1}{n^2}\right)^{\frac{8 n^3}{\pi} \sin \left(\frac{\pi}{2 n}\right)}

is equal to

View answer and explanation

Correct option: B

e^4

Option B (

e^4

) is correct because by manipulating the limit into the form of

e

, the expression simplifies to

e^4

201 Medium

201 of 267

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}

202 Medium

202 of 267

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.

203 Medium

203 of 267

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.

204 Medium

204 of 267

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.

205 Medium

205 of 267

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.

206 Medium

206 of 267

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.

207 Medium

207 of 267

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.

208 Medium

208 of 267

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

209 Medium

209 of 267

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.

210 Medium

210 of 267

Application +4 | -1

\lim \limits_{{n \rightarrow \infty}}\left(\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+\ldots+\frac{1}{n+n}\right)=

View answer and explanation

Correct option: A

\log _e^2

Option A (

\log_e 2

) is correct because the given limit can be expressed as a definite integral, which evaluates to

\log_e 2

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