Let $\alpha $ and $\beta $ be the roots of $a\;{x^2} + b\;x + c = 0$, then $\mathop {\lim }\limits_{x \to a} \frac{{1 - \cos \left( {a\;{x^2} + bx + c} \right)}}{{{{\left( {x - \alpha } \right)}^2}}}$ is equal to

$\frac{1}{2}{\left( {\alpha - \beta } \right)^2}$

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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 31-60 of 930 questions

31 Easy

31 of 930

Analyzing +4 | -1

The value of

\lim\limits_{n \to \infty } \,6\tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}} \left( {\frac{1}{{{r^2} + 3r + 3}}} \right)} \right\}

is equal to:

View answer and explanation

Correct option: C

Option C (3) is correct because the limit is evaluated using the arctangent subtraction formula and telescoping series.

32 Medium

32 of 930

Application +4 | -1

The value of the limit

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

View answer and explanation

Correct option: C

-3

Option C (-3) is correct because L'Hopital's rule is applied after substituting to get the 0/0 form.

33 Medium

33 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 1}} \frac{\int\limits_{1}^x|t-1| d t}{\sin (x-1)}

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct. By applying L'Hopital's rule (or Leibnitz's rule) and evaluating the limit, we find that the limit is 0.

34 Easy

34 of 930

Analyzing +4 | -1

\lim _{x \rightarrow 1^{-}} \frac{\sqrt{\pi}-\sqrt{2 \sin ^{-1} x}}{\sqrt{1-x}}

is equal to:

View answer and explanation

Correct option: B

\sqrt{\frac{2}{\pi}}

Option B (

\sqrt{\frac{2}{\pi}}

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

\sqrt{\frac{2}{\pi}}

35 Medium

35 of 930

Analyzing +4 | -1

Let

{a}_m

be the

{m}^{\text {th }}

term of the sequence

\left(\frac{2^2}{1^2}-\frac{2}{1}\right)^{-1},\left(\frac{3^3}{2^3}-\frac{3}{2}\right)^{-2},\left(\frac{4^4}{3^4}-\frac{4}{3}\right)^{-3}, \ldots .

. , then the value of

\lim \limits_{{m \rightarrow \infty}}\left(a_m\right)^{\frac{1}{m}}

is equal to

View answer and explanation

Correct option: C

\frac{1}{{e}-1}

Option C,

\frac{1}{{e}-1}

, is correct because the limit of the given expression as m approaches infinity simplifies to this value using properties of limits and the definition of e.

36 Easy

36 of 930

Analyzing +4 | -1

\lim \limits_{{x \rightarrow \pi / 2}} \frac{(1-\tan x / 2)(1-\sin x)}{(1+\tan x / 2)(\pi-2 x)^3}

View answer and explanation

Correct option: D

\frac{1}{32}

Option D (1/32) is correct because the limit is evaluated using trigonometric identities and substitution, resulting in 1/32.

37 Medium

37 of 930

Application +4 | -1

\lim\limits_{x \to 0} {\left\{ {\tan \left( {\frac{\pi }{4} + x} \right)} \right\}^{\frac{1}{x}}} =

View answer and explanation

Correct option: B

e^2

Option B (

e^2

) is correct because the limit can be evaluated by using the tangent addition formula and then applying the standard limit formula.

38 Medium

38 of 930

Application +4 | -1

f(1)=1

and

f^{\prime}(1)=4

, then the value of

\lim \limits_{{x \rightarrow 1}} \frac{\sqrt{f(x)}-1}{\sqrt{x}-1}

View answer and explanation

Correct option: A

Option A (4) is correct because it is the limit calculated using L'Hopital's rule after recognizing the indeterminate form.

39 Easy

39 of 930

Analyzing +4 | -1

\lim _{x \rightarrow 0} \frac{(1-\cos 2 x)(3+\cos x)}{x \tan 4 x}

is equal to

View answer and explanation

Correct option: D

Option D (2) is correct because the limit can be evaluated using trigonometric identities and standard limits.

40 Easy

40 of 930

Application +4 | -1

l\left( x \right)

is the least integer not less than

x

and

g\left( x \right)

is the greatest integer not greater than

x

, then

\lim\limits_{x \to e + \pi } \left\{ {l\left( x \right) + g\left( x \right)} \right\}

is equal to

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the limit evaluates to 11, which is not among the provided options.

41 Medium

41 of 930

Analyzing +4 | -1

Let

\alpha

and

\beta

be the roots of

a\;{x^2} + b\;x + c = 0

, then

\lim\limits_{x \to a} \frac{{1 - \cos \left( {a\;{x^2} + bx + c} \right)}}{{{{\left( {x - \alpha } \right)}^2}}}

is equal to

View answer and explanation

Correct option: C

\frac{{{\alpha ^2}}}{2}{\left( {\alpha - \beta } \right)^2}

Option C (

\frac{{{{\alpha ^2}}}}{2}{\left( {\alpha - \beta } \right)^2}

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

42 Easy

42 of 930

Analyzing +4 | -1

Let

f(x)=\frac{\log _e\left(x^2+e^x\right)}{\log _e\left(x^4+e^{2 x}\right)}

. If

\lim \limits_{{x \rightarrow \infty}} f(x)=l

and

\lim \limits_{{x \rightarrow-\infty}} f(x)=m

, then

View answer and explanation

Correct option: A

l=m

Option B (

\sqrt{pq}

) is correct because it is obtained by converting the limit to an exponential form and using L'Hopital's rule or the properties of limits.

43 Easy

43 of 930

Analyzing +4 | -1

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

, is

View answer and explanation

Correct option: D

-2

Option D (-2) is correct because the limit is evaluated using trigonometric identities and simplification, resulting in -2.

44 Easy

44 of 930

Analyzing +4 | -1

\lim _{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 and simplifying, the limit exists and equals

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

45 Easy

45 of 930

Analyzing +4 | -1

\lim\limits_{x \to 0} \frac{{(1 - \cos 2x)(3 + \cos x)}}{{x\tan 4x}}

is equal to

View answer and explanation

Correct option: D

Option D (2) is correct because by using trigonometric identities and limit properties, the given limit simplifies to 2.

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46 Medium

46 of 930

Application +4 | -1

f(9)=9, f^{\prime}(9)=4

, then

\lim \limits_{{x \rightarrow 9}} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3}=

View answer and explanation

Correct option: B

Option B (4) is correct because by applying L'Hôpital's rule, the limit simplifies to 4.

47 Easy

47 of 930

Analyzing +4 | -1

\{x\}

denotes the fractional part of

x

, then

\lim \limits_{{x \rightarrow 1}} \frac{x \sin \{x\}}{x-1}

, is

View answer and explanation

Correct option: C

Non-existent

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

48 Easy

48 of 930

Analyzing +4 | -1

\lim \limits_{{n \rightarrow \infty}}\left(\frac{\sqrt[n]{a}+\sqrt[n]{b}}{2}\right)^n, a, b > 0

equals

View answer and explanation

Correct option: B

\sqrt{a b}

Option B (

\sqrt{ab}

) is correct because the limit can be evaluated by using the exponential form and properties of logarithms.

49 Medium

49 of 930

Analyzing +4 | -1

f(x)=\lim \limits_{{n \rightarrow \infty}} \sum\limits_{{r=0}}^n \frac{\tan \frac{x}{2^{r+1}}+\tan ^3 \frac{x}{2^{r+1}}}{1-\tan ^2 \frac{x}{2^{r+1}}}

then

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

is equal to

View answer and explanation

Correct option: A

Option A (1) is correct because after simplifying the summation and using the limit definition of tan(x)/x as x approaches 0, the limit evaluates to 1.

50 Easy

50 of 930

Analyzing +4 | -1

Let

f(x)=x^3\left\{\sqrt{x^2+\sqrt{x^4+1}}-x \sqrt{2}\right\}

. Then

\lim _{x \rightarrow \infty} f(x)

is equal to

View answer and explanation

Correct option: B

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

Option B (1/(4√2)) is correct because after simplifying the expression and taking the limit as x approaches infinity, the result is 1/(4√2).

51 Easy

51 of 930

Analyzing +4 | -1

\lim\limits_{x \to \infty } \,\,\left[ {\sqrt {{x^2} + 2x - 1} - x} \right]

is equal to

View answer and explanation

Correct option: D

Option D (1) is correct because, by rationalizing the expression and taking the limit as x approaches infinity, we get 1.

52 Easy

52 of 930

Analyzing +4 | -1

f(x)=3 x^{10}-7 x^8+5 x^6-21 x^3+3 x^2-7

, then the value of

\lim \limits_{{h \rightarrow 0}} \frac{f(1-h)-f(1)}{h^3+3 h}

View answer and explanation

Correct option: D

None of these

The correct answer is None of these because the limit evaluates to -53/3, which is not among the given options.

53 Easy

53 of 930

Analyzing +4 | -1

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

View answer and explanation

Correct option: B

is equal to 1

Option B (is equal to 1) is correct because after substituting

t = \frac{{\sqrt{{1 + x^2 + x^4}} - 1}}{{x}}

, the limit simplifies to the standard limit

\lim_{{t \rightarrow 0}} \frac{{e^t - 1}}{{t}} = 1

54 Easy

54 of 930

Analyzing +4 | -1

Let

p(x)

be a real polynomial of degree 4 having extreme values at

x=1

and

x=2

. If

\lim \limits_{{x \rightarrow 0}} \frac{p(x)}{x^2}=1

, then

p(4)

is equal to

View answer and explanation

Correct option: B

Option B (16) is correct because after finding the polynomial p(x) using the given conditions, substituting x=4 yields the correct value.

55 Medium

55 of 930

Application +4 | -1

Let

\alpha=\lim \limits_{{n \rightarrow \infty}} \frac{\left(1^3-1^2\right)+\left(2^3-2^2\right)+\ldots+\left(n^3-n^2\right)}{n^4}

, then

\alpha

is equal to:

View answer and explanation

Correct option: B

\frac{1}{4}

Option B (1/4) is correct because the limit of the given expression as n approaches infinity simplifies to 1/4 after evaluating the sums.

56 Easy

56 of 930

Understanding +4 | -1

\lim\limits_{x \to \infty } \frac{{4{x^5} + 9x + 7}}{{3{x^5} + {x^4} + {x^2} + 1}} =

View answer and explanation

Correct option: C

\frac{4}{3}

Option C (4/3) is correct because as x approaches infinity, the highest power terms dominate, and the limit is the ratio of their coefficients.

57 Medium

57 of 930

Application +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}}

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 and simplifying the expression.

58 Medium

58 of 930

Application +4 | -1

The value of the limit

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

View answer and explanation

Correct option: D

Option D (1) is correct because by substituting and using standard limits, the expression simplifies to 1.

59 Easy

59 of 930

Application +4 | -1

The value of

\lim _{x \rightarrow-1} \frac{x^9+1}{x^{14}+1}

View answer and explanation

Correct option: B

-\frac{9}{14}

Option B (

-\frac{9}{14}

) is correct because L'Hôpital's rule is applied to the limit, resulting in the given value.

60 Easy

60 of 930

Application +4 | -1

f(x)=\left\{\begin{array}{ll}|x|, & x \leq 2 \\ {[x],} & x>2\end{array}\right.

, then .....

View answer and explanation

Correct option: C

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

Option D is correct because the limit from the left and right are not equal, therefore the limit does not exist.

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