Let $f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {{x^2}}&{{\text{if}}}&{x 3} \end{array}} \right.$ Which one of the following is not correct?

$\lim _{x \rightarrow 2} f(x)=4$

$\lim _{x \rightarrow 3} f(x)=2$

$\lim _{x \rightarrow 4} f(x)=4$

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JEE • MATHEMATICS • LIMITS

Limits practice · Mixed difficulty

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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.

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Showing 871-900 of 930 questions

871 Easy

871 of 930

Application +4 | -1

\lim \limits_{{x \rightarrow 2}} \frac{\sum\limits_{{r=1}}^n x^r-\sum\limits_{{r=1}}^n 2^r}{x-2}

is equal to

View answer and explanation

Correct option: C

(n-1) 2^n+1

Option C,

(n-1) 2^n+1

, is correct because it is the result of differentiating the sums, applying the limit, and then simplifying the resulting series.

872 Easy

872 of 930

Understanding +4 | -1

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

, us

View answer and explanation

Correct option: B

Option B (0) is correct because x² approaches 0, and sin(π/x) is bounded between -1 and 1, so their product approaches 0.

873 Easy

873 of 930

Application +4 | -1

The value of

\lim \limits_{{x \rightarrow-1}} \frac{x^2+3 x+2}{x^2+4 x+3}

View answer and explanation

Correct option: D

1 / 2

Option D (

1 / 2

) is correct because the limit is found by factoring and simplifying the rational expression before evaluating at x = -1.

874 Medium

874 of 930

Application +4 | -1

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

View answer and explanation

Correct option: B

Option B (1) is correct because by applying L'Hopital's rule to the logarithmic form of the limit, we find the limit equals 1.

875 Easy

875 of 930

Application +4 | -1

\lim \limits_{{x \rightarrow 0}}\left(1+\frac{a \sin b x}{\cos x}\right)^{\frac{1}{x}}

, where

a, b

are non-zero constants is equal to:

View answer and explanation

Correct option: C

e^{a b}

Option C (e^(ab)) is correct because the limit is of the form 1^infinity, which simplifies to e^(ab) using the standard limit formula.

876 Easy

876 of 930

Application +4 | -1

\lim\limits_{x \to \pi /4} \frac{{1 - {{\cot }^3}x}}{{2 - \cot x - {{\cot }^3}x}}

, is

View answer and explanation

Correct option: B

\frac{3}{4}

Option B (

\frac{3}{4}

) is correct because it is the result of factoring and simplifying the expression after substituting

y = \cot x

and then evaluating the limit.

877 Easy

877 of 930

Analyzing +4 | -1

\lim\limits_{x \to 0} \frac{{3x + \left| x \right|}}{{7x - \left| x \right|}}

, is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the left-hand limit and the right-hand limit are not equal, indicating the limit does not exist.

878 Medium

878 of 930

Analyzing +4 | -1

The value of

\lim _{x \rightarrow 0} \frac{1}{x^3} \int_0^x \frac{t \ln (1+t)}{t^4+4} d t

View answer and explanation

Correct option: B

\frac{1}{12}

Option B (

\frac{1}{12}

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

\frac{1}{12}

879 Easy

879 of 930

Analyzing +4 | -1

\lim \limits_{{y \rightarrow 0}} \frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}

View answer and explanation

Correct option: A

Exists and equals

\frac{1}{4 \sqrt{2}}

Option A (Exists and equals

\frac{1}{4 \sqrt{2}}

) is correct because after rationalizing the numerator twice, the limit can be evaluated to

\frac{1}{4 \sqrt{2}}

880 Medium

880 of 930

Application +4 | -1

Suppose

{f}({x})= \begin{cases}a+b x, & x < 1 \\ 4, & x=1 \\ b-a x, & x>1\end{cases}

and if

\lim \limits_{{x \rightarrow 1}} f(x)=f(1)

then the value of

a+b

View answer and explanation

Correct option: A

Option A is correct because the limit condition implies that a + b = 4.

881 Easy

881 of 930

Application +4 | -1

Let

f: {R} \rightarrow {R}

be a continuous function such that

f(3 x)-f(x)=x

. If

f(8)=7

, then

f(14)

is equal to

View answer and explanation

Correct option: B

Option B (10) is correct because by using the given functional equation and the limit concept, we can find f(0) and then f(14).

882 Medium

882 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow \pi / 2}}\left\{1^{\sec ^2 x}+2^{\sec ^2 x}+3^{\sec ^2 x}+\ldots .+n^{\sec ^2 x}\right\}^{\cos ^2 x},

View answer and explanation

Correct option: B

n

Option B (

n

) is correct because, as

x

approaches

\pi/2

, the expression simplifies to

n

after applying limit properties and L'Hôpital's rule implicitly.

883 Easy

883 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 0}} \frac{2 a x+(a-1) \sin x}{\tan ^3 x}=l

, then

a+l

is equal to

View answer and explanation

Correct option: D

\frac{4}{9}

Option D (4/9) is correct because by applying L'Hopital's rule and standard limits, we find

a = \frac{{1}}{{3}}

and

l = \frac{{1}}{{9}}

, thus

a + l = \frac{{4}}{{9}}

884 Easy

884 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 2}} \frac{\sqrt{x+7}-3 \sqrt{2 x-3}}{\sqrt[3]{x+6}-2 \sqrt[3]{3 x-5}}

View answer and explanation

Correct option: B

\frac{34}{23}

Option B (

\frac{34}{23}

) is correct because the limit is evaluated using L'Hôpital's Rule or by rationalizing and simplifying the expression.

885 Easy

885 of 930

Application +4 | -1

\lim _{n \rightarrow \infty}\left\{\frac{1+\frac{1}{3}+\frac{1}{9}+\cdots \text { to }(n+1) \text { terms }}{1+\frac{1}{2}+\frac{1}{4}+\cdots \text { to }(n+1) \text { terms }}\right\}=

View answer and explanation

Correct option: D

\frac{2}{3}

Option A (2/3) is correct because the limit simplifies to 2/3 after evaluating the geometric series and taking the limit as n approaches infinity.

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886 Easy

886 of 930

Application +4 | -1

The value of

\lim \limits_{{x \rightarrow 2}} \frac{e^{3 x-6}-1}{\sin (2-x)}

, is

View answer and explanation

Correct option: C

-3

Option C (-3) is correct because the limit can be evaluated using the standard limit

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

and

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

887 Easy

887 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 0}} \frac{\log _e(1+x)-x}{x\left\{(1+x)^{1 / x}-e\right\}}

equals to

View answer and explanation

Correct option: C

1 / e

Option C (

\frac{1}{e}

) is correct because by using Taylor series expansion for

\log_e(1+x)

and

(1+x)^{1/x}

, we can simplify the limit and find its value.

888 Easy

888 of 930

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

e

Option B (

e

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

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

and then evaluated.

889 Easy

889 of 930

Application +4 | -1

\lim \limits_{{x \rightarrow \pi / 2}}(\sec x-\tan x)

is equal to

View answer and explanation

Correct option: A

Option A is correct because the limit of (sec x - tan x) as x approaches π/2 is 0.

890 Easy

890 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 0}} \frac{\sqrt{\frac{1}{2}(1-\cos 2 x)}}{x}

, is

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.

891 Easy

891 of 930

Analyzing +4 | -1

Let

f(x)=x(-1)^{[1 / x]}, x \neq 0

, where

[x]

denotes the greatest integer less than or equal to

x

, then

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

View answer and explanation

Correct option: C

Is equal to 0

Option C (Is equal to 0) is correct because as x approaches 0, the function is bounded by x, making the limit 0.

892 Medium

892 of 930

Analyzing +4 | -1

\lim _{\lambda \rightarrow 0}\left(\int_0^1(1+x)^\lambda d x\right)^{1 / \lambda}

is equal to

View answer and explanation

Correct option: B

\frac{4}{e}

Option B (4/e) is correct because, after evaluating the integral and applying L'Hopital's rule, the limit is found to be 4/e.

893 Easy

893 of 930

Analyzing +4 | -1

f\left( x \right)

is differentiable and strictly increasing function, then the value of

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

View answer and explanation

Correct option: C

- 1

Option C (-1) is correct because applying L'Hopital's rule and evaluating the limit results in -1.

894 Easy

894 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{n \rightarrow \infty}}\left(\sqrt{n^2+n+1}-\left[\sqrt{n^2+n+1}\right]\right)

, where [.] denotes the greatest integer function is

View answer and explanation

Correct option: B

1 / 2

Option B (1/2) is correct because the expression simplifies to

\frac{1}{2}

after rationalizing and taking the limit as n approaches infinity.

895 Medium

895 of 930

Understanding +4 | -1

The right hand limit of the function

f(x) = \left\{ {\begin{array}{llllllllllllllllllll} {\frac{{|x - 4|}}{{x - 4}},}&{x \ne 4}\\ {0,}&{x = 4} \end{array}} \right.

x=4

, is

View answer and explanation

Correct option: A

Option A (1) is correct because the right-hand limit is calculated by considering values of x approaching 4 from the right, where the absolute value simplifies to the expression itself.

896 Medium

896 of 930

Analyzing +4 | -1

\lim _{x \rightarrow 2}\left(\sum_{n=1}^9 \frac{x}{n(n+1) x^2+2(2 n+1) x+4}\right)

is equal to

View answer and explanation

Correct option: A

\frac{9}{44}

Option A (

\frac{9}{44}

) is correct because it is the result of evaluating the limit of the summation by using partial fraction decomposition.

897 Medium

897 of 930

Understanding +4 | -1

Let

f(x)=\cos [x]

where

[x]

is the integral part of

x

. Then

View answer and explanation

Correct option: B

\lim _{x \rightarrow 0+0} f(x)=1

Option B is correct because as x approaches 0 from the right, [x] = 0, so cos([x]) = cos(0) = 1.

898 Easy

898 of 930

Analyzing +4 | -1

{S_n} = \sum \limits_{k = 1}^n \,\,{a_k}

and

\lim\limits_{n \to \infty } {a_n} = a

, then

\lim\limits_{n \to \infty } \frac{{{S_{n + 1}} - {S_n}}}{{\sqrt {\sum \nolimits_{k = 1}^n \,k} }}

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because the limit of the given expression as n approaches infinity is 0.

899 Easy

899 of 930

Application +4 | -1

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

, where [.] denotes the greatest integer function, is

View answer and explanation

Correct option: B

\frac{8}{3}

Option B (

5 \sqrt{3}

) is correct because the area of a parallelogram is half the magnitude of the cross product of its diagonals.

900 Easy

900 of 930

Analyzing +4 | -1

Let

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {{x^2}}&{{\text{if}}}&{x < 1} \\ {x + 2}&{{\text{if}}}&{1 \leqslant x \leqslant 2} \\ {8 - 2x}&{{\text{if}}}&{2 < x \leqslant 3} \\ {2x - 4}&{i{\text{f}}}&{x > 3} \end{array}} \right.

Which one of the following is not correct?

View answer and explanation

Correct option: A

\lim _{x \rightarrow 1} f(x)=1

Option A is correct because the left-hand limit at x=1 is 1, but the right-hand limit is 3, therefore the limit does not exist, making the statement incorrect.

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