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JEE Main Limits, Continuity and Differentiability Practice Questions with Solutions

Limits, Continuity and Differentiability is an important topic from the Calculus part of Mathematics.. It is also a scoring section, so students should make the most out of it through regular practice. The JEE Main Limits, Continuity and Differentiability Practice Test is a valuable resource for all JEE Main aspirants to learn about important mathematical concepts and enhance their problem-solving abilities. The JEE Main Limits, Continuity and Differentiability Practice Questions with Solutions have many different questions from the Sequences and Series chapter for students to practice and strengthen their mathematics section for the JEE Main entrance exam.

About 2 to 4 questions may appear from Limits, Continuity and Differentiability on the JEE Main question paper. You will get multiple-choice questions and integer-type questions in the question paper, which will test your problem-solving abilities. JEE Main is a challenging entrance exam for all engineering aspirants. Therefore, it is advised for candidates to attempt these practice tests at least once a week to develop knowledge and precision of the topic.

Around 15 to 20 questions are available from this chapter in the JEE Main Limits, Continuity and Differentiability Practice Questions with Solutions, covering all the essential topics. The important topics of this chapter, from which questions are likely to appear in the exam, are Limits, Continuity, Differentiability, Mean Value Theorems, Differentiation Techniques, and Graphical & Conceptual Problems. The practice tests will contain questions on all these topics, along with their detailed solutions for students to practice. Students may go through the solutions and their explanations provided for their understanding.

Practising the JEE Main Limits, Continuity and Differentiability Practice Questions with Solutions on a regular basis will help the students to improve their accuracy in solving problems and manage their time better. It will also allow them to recall formulas, understand concepts, and solve questions from this topic easily in the JEE Main exam.

JEE Main Mathematics Limits, Continuity and Differentiability Practice Questions

Chemistry Physics

Question 1.

$lim_{x \to 0} \frac{e - (1 + 2 x)^{\frac{1}{2 x}}}{x}$ is equal to

\frac{- 2}{e}

e - e^{2}

e

Question 2.

For $a, b > 0$ , let $f (x) = {\begin{cases} \frac{\tan ((a + 1) x) + b \tan x}{x}, & x < 0 \\ 3, & x = 0 \\ \frac{\sqrt{a x + b^{2} x^{2}} - \sqrt{a x}}{b \sqrt{a} x \sqrt{x}}, & x > 0 \end{cases}$ be a continuous function at $x = 0$ . Then $\frac{b}{a}$ is equal to :

Question 3.

$lim_{n \to \infty} \frac{(1^{2} - 1) (n - 1) + (2^{2} - 2) (n - 2) + \dots + ((n - 1)^{2} - (n - 1)) \cdot 1}{(1^{3} + 2^{3} + \dots \dots + n^{3}) - (1^{2} + 2^{2} + \dots \dots + n^{2})}$ is equal to :

\frac{2}{3}

\frac{1}{2}

\frac{3}{4}

\frac{1}{3}

Question 4.

Let , $f : [- 1, 2] \to R$ be given by $f (x) = 2 x^{2} + x + [x^{2}] - [x]$ , where $[t]$ denotes the greatest integer less than or equal to $t$ . The number of points, where $f$ is not continuous, is :

Question 5.

If the function $f (x) = \frac{\sin 3 x + α \sin x - β \cos 3 x}{x^{3}}, x \in R$ , is continuous at $x = 0$ , then $f (0)$ is equal to :

-

-

Question 6.

If the function

$f (x) = {\begin{cases} \frac{72^{x} - 9^{x} - 8^{x} + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ a \log_{e} 2 \log_{e} 3 & , x = 0 \end{cases}$

is continuous at $x = 0$ , then the value of $a^{2}$ is equal to

968

1250

1152

746

Question 7.

Let $f : R \to R$ be a function given by

$f (x) = {\begin{cases} \frac{1 - \cos 2 x}{x^{2}}, & x < 0 \\ α, & x = 0, \\ \frac{β \sqrt{1 - \cos x}}{x}, & x > 0 \end{cases}$

where $α, β \in R$ . If $f$ is continuous at $x = 0$ , then $α^{2} + β^{2}$ is equal to :

Question 8.

Let

f (x) = | 2 x^{2} + 5 | x | - 3 |, x \in R

. If

m

and

n

denote the number of points where

f

is not continuous and not differentiable respectively, then

m + n

is equal to :

Question 9.

Let

f (x) = {\begin{cases} x - 1, x is even, \\ 2 x, x is odd, \end{cases} x \in N

.

If for some

a \in N, f (f (f (a))) = 21

, then

lim_{x \to a^{-}} {\frac{| x |^{3}}{a} - [\frac{x}{a}]}

, where

[t]

denotes the greatest integer less than or equal to

t

, is equal to :

169

121

225

144

Question 10.

Let

f : R \to R

be defined as :

f (x) = {\begin{cases} \frac{a - b \cos 2 x}{x^{2}}; & x < 0 \\ x^{2} + c x + 2; & 0 \leq x \leq 1 \\ 2 x + 1; & x > 1 \end{cases}

f

is continuous everywhere in

R

and

m

is the number of points where

f

is NOT differential then

m + a + b + c

equals :

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Question 1.

Consider the function $f : (0, \infty) \to R$ defined by $f (x) = e^{- | \log_{e} x |}$ . If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m + n$ is

Question 2.

$lim_{x \to 0} \frac{e^{2 | \sin x |} - 2 | \sin x | - 1}{x^{2}}$

is equal to 1

does not exist

is equal to

- 1

is equal to 2

Question 3.

Let $g (x)$ be a linear function and $f (x) = {\begin{array}{cl} g (x) & , x \leq 0 \\ {(\frac{1 + x}{2 + x})}^{\frac{1}{x}} & , x > 0 \end{array}$ , is continuous at $x = 0$ . If $f^{'} (1) = f (- 1)$ , then the value $g (3)$ is

\log_{e} (\frac{4}{9}) - 1

\frac{1}{3} \log_{e} (\frac{4}{9 e^{1 / 3}})

\log_{e} (\frac{4}{9 e^{1 / 3}})

\frac{1}{3} \log_{e} (\frac{4}{9}) + 1

Question 4.

Consider the function $f : (0, 2) \to R$ defined by $f (x) = \frac{x}{2} + \frac{2}{x}$ and the function $g (x)$ defined by

$g (x) = {\begin{cases} min ⌊ f (t)}, & 0 < t \leq x and 0 < x \leq 1 \\ \frac{3}{2} + x, & 1 < x < 2 \end{cases} . Then,$

g

is continuous but not differentiable at

x = 1

g

is continuous and differentiable for all

x \in (0, 2)

g

is not continuous for all

x \in (0, 2)

g

is neither continuous nor differentiable at

x = 1

Question 5.

$If lim_{x \to 0} \frac{3 + α \sin x + β \cos x + \log_{e} (1 - x)}{3 \tan^{2} x} = \frac{1}{3}, then 2 α - β is equal to :$

Question 6.

Consider the function.

f (x) = {\begin{array}{cc} \frac{a (7 x - 12 - x^{2})}{b | x^{2} - 7 x + 12 |} & , x < 3 \\ 2^{\frac{\sin (x - 3)}{x - [x]}} & , x > 3 \\ b & , x = 3, \end{array}

where

[x]

denotes the greatest integer less than or equal to

x

. If

S

denotes the set of all ordered pairs (a, b) such that

f (x)

is continuous at

x = 3

, then the number of elements in

S

is :

Infinitely many

Question 7.

a = lim_{x \to 0} \frac{\sqrt{1 + \sqrt{1 + x^{4}}} - \sqrt{2}}{x^{4}}

and

b = lim_{x \to 0} \frac{\sin^{2} x}{\sqrt{2} - \sqrt{1 + \cos x}}

, then the value of

a b^{3}

is :

Question 8.

Let

[x]

denote the greatest integer function and

f (x) = max {1 + x + [x], 2 + x, x + 2 [x]}, 0 \leq x \leq 2

. Let

m

be the number of

points in

[0, 2]

, where

f

is not continuous and

n

be the number of points in

(0, 2)

, where

f

is not differentiable. Then

(m + n)^{2} + 2

is equal to :

Question 9.

If $lim_{x \to 0} \frac{e^{a x} - \cos (b x) - \frac{c x e^{- c x}}{2}}{1 - \cos (2 x)} = 17$ , then $5 a^{2} + b^{2}$ is equal to

Question 10.

Let $f$ and $g$ be two functions defined by

$f (x) = {\begin{array}{cc} x + 1, & x < 0 \\ | x - 1 |, & x \geq 0 \end{array}$ and $g (x) = {\begin{array}{cc} x + 1, & x < 0 \\ 1, & x \geq 0 \end{array}$

Then $(g \circ f) (x)$ is :

continuous everywhere but not differentiable at

x = 1

differentiable everywhere

not continuous at

x = - 1

continuous everywhere but not differentiable exactly at one point

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JEE Main Limits, Continuity and Differentiability Practice Questions With Solutions