$\mathop {\lim }\limits_{x \to 1} \frac{{\sqrt {1 - \cos 2\left( {x - 1} \right)} }}{{x - 1}}$

Exists and is equals $\sqrt 2 $

Exists and is equals $ - \sqrt 2 $

Does not exist because $x - 1 \to 0$

If $f(x)=\left\{\begin{array}{l}\frac{5}{2}-x, \text { when } x 2\end{array}\right.$, then:

$\lim _{x \rightarrow 2} f(x)=1$

Back to LIMITS

Your Progress

0 / 647 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • LIMITS

Limits practice · Easy

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

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.

Take a chapter quiz →

Showing 541-570 of 647 questions

Preparing practice and saved progress…

541 Easy

541 of 647

Analyzing +4 | -1

\lim\limits_{x \to a} \frac{{{a^x} - {x^a}}}{{{x^x} - {a^a}}} = - 1

, then

a

equal to

View answer and explanation

Correct option: A

Option A (1) is correct because by applying L'Hospital's rule, the value of 'a' is found to be 1.

542 Easy

542 of 647

Application +4 | -1

The value of

\lim \limits_{{n \rightarrow \infty}} \frac{1 \cdot 2+2 \cdot 3+3 \cdot 4+\ldots+n(n+1)}{n^3}

30, is

View answer and explanation

Correct option: C

\frac{1}{3}

Option C (

\frac{1}{3}

) is correct because the sum in the numerator simplifies to

\frac{n(n+1)(n+2)}{3}

, and dividing by

n^3

and taking the limit as

n

approaches infinity yields

\frac{1}{3}

543 Easy

543 of 647

Understanding +4 | -1

f(x)=\left\{\begin{array}{rr}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{array}\right.

, then

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

View answer and explanation

Correct option: B

Option B (0) is correct because the limit of the function as x approaches 0 is 0, as shown by the Squeeze Theorem.

544 Easy

544 of 647

Analyzing +4 | -1

\alpha

and

\beta

are roots of

x^2-(\sqrt{1-\cos 2 \theta}) x+\theta=0

, where

0<\theta<\frac{\pi}{2}

. Then,

\lim \limits_{\theta \rightarrow 0^{+}}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)

View answer and explanation

Correct option: C

\sqrt{2}

The correct answer is C (

\sqrt{2}

) because by simplifying the expression and using the limit definition, the value can be determined as

\sqrt{2}

545 Easy

545 of 647

Application +4 | -1

\lim \limits_{{x \rightarrow-a}} \frac{x^9+a^9}{x+a}=9

, then

a=

View answer and explanation

Correct option: A

\pm 1

Option C (1/2) is correct because simplifying the expression using the sum of arithmetic series leads to a limit of 1/2 as n approaches infinity.

546 Easy

546 of 647

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 1}} \frac{a \sin (x-1)+b \cos (x-1)+4}{x^2-1}=2

, then

(a, b)

is equal to

View answer and explanation

Correct option: D

(4,-4)

Option B (3x+4y-5=0) is correct because the values of a and b obtained from the limit condition satisfy this equation.

547 Easy

547 of 647

Analyzing +4 | -1

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

, then

View answer and explanation

Correct option: C

a \in(-\infty, 1)

Option C (

a \in(-\infty, 1)

) is correct because for the limit to be infinite, the degree of numerator must be greater than the degree of the denominator, implying

1-a > 0

548 Easy

548 of 647

Application +4 | -1

\lim _{x \rightarrow 0}\left(\frac{1+6 x^2}{1+3 x^2}\right)^{1 / x^2}=

View answer and explanation

Correct option: C

e^3

Option C (

e^3

) is correct because it is obtained by applying the limit formula

\lim_{x \rightarrow 0} (1 + ax)^{1/x} = e^a

to both the numerator and denominator.

549 Easy

549 of 647

Application +4 | -1

The value of

\lim\limits_{x \to \frac{\pi }{2}} {\left( {\sin x} \right)^{\tan x}}

View answer and explanation

Correct option: A

Option A (1) is correct because by taking the logarithm and applying L'Hopital's rule, the limit simplifies to 1.

550 Easy

550 of 647

Application +4 | -1

f(1)=1, f^{\prime}(1)=2

then

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

View answer and explanation

Correct option: A

Option A (2) is correct because the limit can be evaluated using L'Hopital's rule or by algebraic manipulation and the given derivative.

551 Easy

551 of 647

Analyzing +4 | -1

\lim \limits_{{x \rightarrow \infty}}\left\{\frac{x^2+1}{x+1}-a x-b\right\}=2

, then

View answer and explanation

Correct option: D

a=1, b=-3

Option D (

a=1, b=-3

) is correct because for the limit to be 2, the coefficients of

x^2

must be 0 and the ratio of the coefficients of

x

must be 2.

552 Easy

552 of 647

Application +4 | -1

The value of

\lim\limits_{x \to \infty } x\cos \left( {\frac{\pi }{{4x}}} \right)\sin \left( {\frac{\pi }{{4x}}} \right)

, is

View answer and explanation

Correct option: B

\frac{\pi }{4}

Option B (

\frac{\pi }{4}

) is correct because by using the identity

2\sin x \cos x = \sin 2x

and the limit

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

, the given limit simplifies to

\frac{\pi }{4}

553 Easy

553 of 647

Application +4 | -1

\lim \limits_{{x \rightarrow a}} \frac{(a+2 x)^{1 / 3}-(3 x)^{1 / 3}}{(3 a+x)^{1 / 3}-(4 x)^{1 / 3}},(a \neq 0)

is equal to

View answer and explanation

Correct option: D

\frac{2}{3}\left(\frac{2}{9}\right)^{1 / 3}

Option D,

\frac{2}{3}(\frac{2}{9})^{1 / 3}

, is correct because it is the result of applying L'Hopital's rule to the limit and simplifying the expression.

554 Easy

554 of 647

Application +4 | -1

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

, then

k=

View answer and explanation

Correct option: B

\frac{8}{3}

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

555 Easy

555 of 647

Analyzing +4 | -1

\lim\limits_{x \to \infty } \left\{ {\sqrt {{x^2} - x + 1} - ax - b} \right\} = 0

, then

View answer and explanation

Correct option: B

a = 1,\;b = - \frac{1}{2}

Option B (

a = 1, b = -\frac{1}{2}

) is correct because after rationalizing and comparing coefficients, these are the values that make the limit equal to 0.

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

556 Easy

556 of 647

Analyzing +4 | -1

\lim\limits_{x \to 1} \frac{{\sqrt {1 - \cos 2\left( {x - 1} \right)} }}{{x - 1}}

View answer and explanation

Correct option: D

Does not exist because left hand limit is not equal to right hand limit

Option A (

{\left( {\log a} \right)^2}

) is correct. Applying L'Hopital's rule twice gives the result

{\left( {\log a} \right)^2}

557 Easy

557 of 647

Application +4 | -1

The value of

\lim \limits_{{x \rightarrow 0}} \frac{1}{x} \sin ^{-1}\left(\frac{2 x}{1+x^2}\right)

View answer and explanation

Correct option: A

Option A (2) is correct because by using the trigonometric identity and limit properties, the given limit evaluates to 2.

558 Easy

558 of 647

Analyzing +4 | -1

n \in N

, and

[x]

denotes the greatest integer less than or equal to

x

, then

\lim\limits _{x \rightarrow n}(-1)^{[x]}

is equal to

View answer and explanation

Correct option: D

None of these

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

559 Easy

559 of 647

Application +4 | -1

f(x)=\frac{1-\tan x}{4 x-\pi}, x \neq \frac{\pi}{4}, x \in\left[0, \frac{\pi}{2}\right)

is a continuous functions, then

f(\pi / 4)

is equal to -

View answer and explanation

Correct option: A

-\frac{1}{2}

Option A (-

\frac{1}{2}

) is correct because it is the limit of the function as x approaches

\frac{\pi}{4}

, ensuring continuity.

560 Easy

560 of 647

Application +4 | -1

Let

z_n=\frac{1^3+2^3+3^3+\cdots+n^3}{n^4}

Then

\lim _{n \rightarrow \infty} z_n

View answer and explanation

Correct option: C

\frac{1}{4}

Option C (

\frac{1}{4}

) is correct because the sum of the cubes of the first n natural numbers is

\frac{n^2(n+1)^2}{4}

, and dividing by

n^4

and taking the limit as

n

approaches infinity yields

\frac{1}{4}

561 Easy

561 of 647

Application +4 | -1

f(x)=\left\{\begin{array}{ccc}-2 \sin x & \text { if } & x \leq-\frac{\pi}{2} \\ A \sin x+B & \text { if } & -\frac{\pi}{2} < x < \frac{\pi}{2} \\ \cos x & \text { if } & x \geq \frac{\pi}{2}\end{array}\right.

is a continuous function then

View answer and explanation

Correct option: B

A=-1, B=1

Option B (

A=-1, B=1

) is correct because these values ensure the function is continuous at

x=-\frac{\pi}{2}

and

x=\frac{\pi}{2}

562 Easy

562 of 647

Application +4 | -1

\lim \limits_{{x \rightarrow \infty}}\left(1+\frac{a}{x}-\frac{4}{x^2}\right)^{2 x}=e^3

, then '

a

' is equal to

View answer and explanation

Correct option: B

\frac{3}{2}

Option B (

\frac{{3}}{{2}}

) is correct because the limit can be evaluated using the exponential form of limits.

563 Easy

563 of 647

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: A

\frac{1}{8 \sqrt{3}}

Option A (

\frac{1}{8 \sqrt{3}}

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

564 Easy

564 of 647

Application +4 | -1

The value of

\lim \limits_{{x \rightarrow 0}}(\cos a x)^{\operatorname{cosec}^2 b x}

View answer and explanation

Correct option: C

e^{\left(-\frac{a^2}{2 b^2}\right)}

Option C (

e^{\left(-\frac{a^2}{2 b^2}\right)}

) is correct because the limit can be solved by taking the natural logarithm of the expression and applying L'Hopital's rule or using trigonometric identities.

565 Easy

565 of 647

Application +4 | -1

\lim \limits_{{x \rightarrow \infty}}\{\sqrt[3]{(x+a)(x+b)(x+c)}-x\}=

View answer and explanation

Correct option: B

\frac{a+b+c}{3}

Option B,

\frac{a+b+c}{3}

, is correct because it is the result of rationalizing the expression and then taking the limit.

566 Easy

566 of 647

Analyzing +4 | -1

f(x)=\left\{\begin{array}{r}\frac{|x-a|}{x-a}, \text { when } x \neq a \\ 1, \text { when } x=a\end{array}\right.

, then:

View answer and explanation

Correct option: B

f(x)

is discontinuous at

x=a

Option B (

f(x)

is discontinuous at

x=a

) is correct because the left-hand limit and right-hand limit at

x=a

are not equal, indicating a discontinuity.

567 Easy

567 of 647

Analyzing +4 | -1

f(x)=\left\{\begin{array}{l}\frac{5}{2}-x, \text { when } x<2 \\ 1, \text { when } x=2 \\ x-\frac{3}{2}, \text { when } x>2\end{array}\right.

, then:

View answer and explanation

Correct option: B

f(x)

is discontinuous at

x=2

The function f(x) is discontinuous at x=2 because the limit as x approaches 2 is 1/2, while f(2) = 1, thus they are not equal.

568 Easy

568 of 647

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: C

Option C (2) is correct. The limit is evaluated by converting the sum to an integral and then evaluating the integral.

569 Easy

569 of 647

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 0}} x \log _e(\sin x)

is equal to

View answer and explanation

Correct option: B

\log _e 1

Option B (

\log_e 1

) is correct because the limit is evaluated using L'Hospital's Rule, resulting in 0, which is equal to

\log_e 1

570 Easy

570 of 647

Application +4 | -1

If the function

f(x)=\frac{\cos ^2 x-\sin ^2 x-1}{\sqrt{x^2+1}-1}

is continuous at

x=0

then

f(0)

is equal to

View answer and explanation

Correct option: D

-4

Option D (-4) is correct because the limit of the function as x approaches 0 is -4, and for the function to be continuous at x=0, f(0) must equal this limit.

Revision notes and key takeaways

Review LIMITS in your textbook or class notes, then use this set to check your understanding.

Showing 541-570 of 647 questions

Your Progress

Limits practice · Easy

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice