JEE Main Limits, Continuity and Differentiability Practice Questions With Solutions

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JEE Main Mathematics Limits, Continuity and Differentiability Practice Questions

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 :

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

JEE Main Mathematics Limits, Continuity and Differentiability Practice Questions

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

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