$\mathop {\lim }\limits_{x \to \frac{\pi }{4}} \frac{{\mathop \smallint \nolimits_2^{{{\sec }^2}x} f\left( t \right)dt}}{{{x^2} - \frac{{{\pi ^2}}}{{16}}}}$ equals

$\frac{2}{\pi }f\left( 2 \right)$

$\frac{2}{\pi }f\left( {\frac{1}{2}} \right)$

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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 841-870 of 930 questions

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841 Easy

841 of 930

Analyzing +4 | -1

The value of

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

, is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the limit evaluates to 1/2, which is not among the given options.

842 Easy

842 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow \infty}}\left(1+\frac{1}{x^n}\right)^x, n > 0

, is

View answer and explanation

Correct option: B

1, if

n > 1

Option B (1, if

n > 1

) is correct because the limit evaluates to 1 when n > 1, using properties of exponential functions.

843 Easy

843 of 930

Application +4 | -1

The value of

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

View answer and explanation

Correct option: C

- 3

Option C (-3) is correct because by applying L'Hopital's rule, the limit can be evaluated to -3.

844 Medium

844 of 930

Analyzing +4 | -1

\lim \limits_{{n \rightarrow \infty}} \prod\limits_{{r=3}}^n \frac{r^3-8}{r^3+8}

is equal to

View answer and explanation

Correct option: A

\frac{2}{7}

Option A (

\frac{2}{7}

) is correct because after factoring and simplifying the product, the limit equals

\frac{2}{7}

845 Easy

845 of 930

Application +4 | -1

\lim \limits_{{x \rightarrow \infty}}\left(1-\frac{4}{x-1}\right)^{3 x-1}

is ....

View answer and explanation

Correct option: A

e^{-12}

Option A (

e^{-12}

) is correct because by applying the limit definition of e, the expression converges to

e^{-12}

846 Easy

846 of 930

Analyzing +4 | -1

f\left( x \right) = \frac{{\sin \left( {{e^{x - 2}} - 1} \right)}}{{\log \left( {x - 1} \right)}}

, then

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

is given by

View answer and explanation

Correct option: D

Option D (1) is correct. By using algebraic manipulation and standard limits, the limit can be evaluated to 1.

847 Medium

847 of 930

Application +4 | -1

\lim \limits_{{x \rightarrow 0}}\left(\frac{1^x+2^x+3^x+\cdots+n^x}{n}\right)^{a / x}

is equal to

View answer and explanation

Correct option: C

(\underline{n})^{a / n}

Option C ((

\underline{n}

)

^{a / n}

) is correct because the limit is evaluated using the exponential limit formula and properties of logarithms, resulting in (

\underline{n}

)

^{a / n}

848 Easy

848 of 930

Analyzing +4 | -1

f\left( x \right)

is the integral function of the function

\frac{{2\sin x - \sin 2x}}{{{x^3}}},x \ne 0

, then

{\lim _ \to }f'\left( x \right)

is equal to

View answer and explanation

Correct option: B

Option B (1) is correct because it is the result of evaluating the limit of the derivative of the integral function.

849 Easy

849 of 930

Application +4 | -1

\lim\limits_{x \to 1} \frac{{{e^{ - x}} - {e^{ - 1}}}}{{x - 1}}

is equal to

View answer and explanation

Correct option: B

- \frac{1}{e}

Option B (

\frac{-1}{e}

) is correct because the limit is the derivative of

e^{-x}

evaluated at

x=1

, which equals

\frac{-1}{e}

850 Easy

850 of 930

Application +4 | -1

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

is equal to

p \cos {q}

, then sum of

p

and

q

View answer and explanation

Correct option: A

Option A is correct because the limit evaluates to 2cos(2), which means p+q = 2+2 = 4.

851 Easy

851 of 930

Analyzing +4 | -1

\lim\limits_{x \to a} \frac{{\log \left( {x - a} \right)}}{{\log \left( {{e^x} - {e^a}} \right)}}

is equal to

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.

852 Easy

852 of 930

Analyzing +4 | -1

\lim _{x \rightarrow 0} \frac{a e^x-b}{x}=2

then

(a, b)

is equal to

View answer and explanation

Correct option: C

(2,2)

Option C ((2,2)) is correct because by applying the limit definition and L'Hopital's rule, we can determine the values of a and b.

853 Medium

853 of 930

Application +4 | -1

\lim\limits_{x \to \frac{\pi }{4}} \frac{{\int \nolimits_2^{{{\sec }^2}x} f\left( t \right)dt}}{{{x^2} - \frac{{{\pi ^2}}}{{16}}}}

equals

View answer and explanation

Correct option: A

\frac{8}{\pi }f\left( 2 \right)

Option A (

\frac{8}{\pi}f(2)

) is correct because it results from applying L'Hopital's rule and evaluating the derivative of

\sec^2 x

x = \frac{\pi}{4}

854 Easy

854 of 930

Application +4 | -1

The value of

\lim _{x \rightarrow 1} \frac{x+x^2+\cdots+x^n-n}{x-1}

View answer and explanation

Correct option: C

\frac{n(n+1)}{2}

Option C (

\frac{n(n+1)}{2}

) is correct because applying L'Hôpital's rule results in the sum of the first n natural numbers, which is

\frac{n(n+1)}{2}

855 Medium

855 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 0}} \frac{\sin (\alpha x-\ln (1+\beta x))}{e^{3 x}-3 x-1}=1

, then

\alpha+2 \beta

is equal to

View answer and explanation

Correct option: C

Option C (9) is correct because by using L'Hopital's rule and series expansion, we find that

\alpha = \beta

and

\beta^2 = 9

, leading to

\alpha + 2\beta = 9

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856 Easy

856 of 930

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 0}} \frac{(\tan (\{x\}-1)) \sin \{x\}}{\{x\}(\{x\}-1)}

, where

\{x\}

denotes the fractional part function is

View answer and explanation

Correct option: D

Doesn't exist

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

857 Easy

857 of 930

Application +4 | -1

[x]

denotes the greatest integer less than or equal to

x

, then

\lim \limits_{{n \rightarrow \infty}} \frac{[x]+[2 x]+[3 x]+\ldots . .+[n x]}{n^2} \text { equals }

View answer and explanation

Correct option: A

x / 2

Option A (

x/2

) is correct because using the properties of the greatest integer function and the sandwich theorem, the limit evaluates to

x/2

858 Medium

858 of 930

Understanding +4 | -1

The equivalent definition of the function given by

f(x)=\left\{\begin{array}{ll}2 x, & x \geq 0 \\ 0, & x<0\end{array}\right.

View answer and explanation

Correct option: C

f(x)=|x|+x

Option C (

f(x)=|x|+x

) is correct because when

x \geq 0

|x| = x

, so

f(x) = x + x = 2x

, and when

x < 0

|x| = -x

, so

f(x) = -x + x = 0

, matching the given definition.

859 Easy

859 of 930

Application +4 | -1

\lim \limits_{{x \rightarrow 0}}(1+a x)^{b / x}=e^4

, where

a

and

b

are natural numbers then

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the condition ab=4 is derived from the limit, but none of the given options satisfy this condition.

860 Easy

860 of 930

Analyzing +4 | -1

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

equals

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because after applying L'Hopital's rule twice, the limit evaluates to -1/2.

861 Easy

861 of 930

Analyzing +4 | -1

\lim \limits_{{h \rightarrow 0}} \frac{\log _e(1+2 h)-2 \log _e(1+h)}{h^2}

View answer and explanation

Correct option: C

-1

Option C (-1) is correct because after applying L'Hôpital's rule, the limit evaluates to -1.

862 Easy

862 of 930

Application +4 | -1

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

View answer and explanation

Correct option: B

3 / 4

Option B (

3 / 4

) is correct because the expression is simplified by factoring and canceling terms, then the limit is evaluated by direct substitution.

863 Easy

863 of 930

Application +4 | -1

f(x)=\left\{\begin{array}{l}x^2-1,0 < x < 2 \\ 2 x+3,2 \leq x < 3\end{array}\right.

, then

\lim \limits_{x \rightarrow 2^{-}} f(x)

and

\lim \limits_{x \rightarrow 2^{+}} f(x)

are

View answer and explanation

Correct option: A

3, 7

Option A (3, 7) is correct because it represents the left-hand and right-hand limits of the function at x=2, respectively.

864 Easy

864 of 930

Analyzing +4 | -1

\lim\limits_{x \to 0} \left[ {\frac{{8\sin x + x\cos x}}{{3\tan x + {x^2}}}} \right]

is equal to

View answer and explanation

Correct option: A

Option A (3) is correct because the limit can be evaluated by dividing both the numerator and denominator by x and then applying the standard limits.

865 Medium

865 of 930

Application +4 | -1

The value of

\lim \limits_{x \rightarrow \sqrt{2}} \frac{x^4-4}{x^2+3 \sqrt{2} x-8}

View answer and explanation

Correct option: C

\frac{8}{5}

Option C (8/5) is correct because after factoring and simplifying the expression, the limit as x approaches sqrt(2) is 8/5.

866 Medium

866 of 930

Analyzing +4 | -1

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

equals

View answer and explanation

Correct option: A

\frac{1}{2}

Option A (

\frac{1}{2}

) is correct because using Taylor series expansions for cos(x) and log(1+x) allows us to evaluate the limit.

867 Easy

867 of 930

Application +4 | -1

\lim\limits_{x \to 0} \left( {\frac{x}{{\sqrt {1 + x} - \sqrt {1 - x} }}} \right)

is equal to

View answer and explanation

Correct option: B

Option B (1) is correct because by rationalizing the denominator, the limit evaluates to 1.

868 Easy

868 of 930

Analyzing +4 | -1

\lim\limits_{x \to 0} \frac{{x({e^{\left( {\sqrt {1 + {x^2} + {x^4}} - 1)/x} \right)}} - 1}}{{\sqrt {1 + {x^2} + {x^4} - 1} }}

View answer and explanation

Correct option: B

is equal to 1

Option B is correct because by applying limits and L'Hopital's rule, the expression simplifies to 1.

869 Easy

869 of 930

Analyzing +4 | -1

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

, where

n=100

is equal to

View answer and explanation

Correct option: C

-\frac{5050}{\pi e}

Option C (-

\frac{5050}{\pi e}

) is correct because the limit is evaluated using L'Hôpital's Rule and series expansion.

870 Easy

870 of 930

Application +4 | -1

\lim\limits_{x \to \infty } \frac{{\left( {2x - 3} \right)\left( {3x - 4} \right)}}{{\left( {4x - 5} \right)\left( {5x - 6} \right)}}

is equal to

View answer and explanation

Correct option: D

\frac{3}{{10}}

Option D (

\frac{3}{{10}}

) is correct because as x approaches infinity, the limit of the rational function is the ratio of the leading coefficients after applying L'Hopital's rule or dividing by the highest power of x.

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