Core Syllabus Module

Limits, Continuity and Differentiability

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 2020

LEVELJEE Advanced

Let the functions $f : (- 1, 1) \to R$ and $g : (- 1, 1) \to (- 1, 1)$ be defined by $f (x) = ∣2 x - 1∣ + ∣2 x + 1∣$ and $g (x) = x - [x]$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Let $f \circ g : (- 1, 1) \to R$ be the composite function defined by $(f \circ g) (x) = f (g (x))$ . Suppose $c$ is the number of points in the interval $(- 1, 1)$ at which $f \circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(- 1, 1)$ at which $f \circ g$ is NOT differentiable. Then the value of $c + d$ is ____.

JEE Main 2022 (25 July Shift 1)

LEVELJEE Advanced

Let $f (x) = {∣4 x^{2} - 8 x + 5∣, [4 x^{2} - 8 x + 5], if 8 x^{2} - 6 x + 1 \geq 0 if 8 x^{2} - 6 x + 1 < 0$ where $[α]$ denotes the greatest integer less than or equal to $α$ . Then the number of points in $R$ where $f$ is not differentiable is ______.

JEE Main 2025 (January)

LEVELJEE Advanced

Let the function, $f (x) = {- 3 a x^{2} - 2, a^{2} + b x, x < 1 x \geq 1$ be differentiable for all $x \in R$ where $a > 1,$ $b \in R$ . If the area of the region enclosed by $y = f (x)$ and the line $y = - 20$ is $α + β 3,$ $α, β \in Z$ then the value of $α + β$ is

JEE Main 2021 (16 March Shift 2)

LEVELJEE Advanced

Let $α \in R$ be such that the function $f (x) = {\frac{c o s ^{- 1} ( 1 - { x } ^{2} ) s i n ^{- 1} ( 1 - { x })}{{ x } - { x } ^{3}}, α, x \neq = 0 x = 0$ is continuous at $x = 0$ , where ${x} = x - [x]$ , $[x]$ is the greatest integer less than or equal to $x$ . Then :

(A)

α = \frac{π}{2}

(B)

α = 0

(C)

no such

α

exists

(D)

α = \frac{π}{4}

JEE Main 2021 (February)

LEVELJEE Advanced

A function $f$ is defined on $[- 3, 3]$ as $f (x) = {min {∣ x ∣, 2 - x^{2}}, [∣ x ∣], - 2 \leq x \leq 2 2 < ∣ x ∣ \leq 3$ where $[x]$ denotes the greatest integer $\leq x$ . The number of points, where $f$ is not differentiable in $(- 3, 3)$ is

JEE(ADVANCED)-201

LEVELJEE Advanced

Let $[x]$ be the greatest integer less than or equals to $x$ . Then, at which of the following point(s) the function $f (x) = x cos (π (x + [x]))$ is discontinuous ?

* Multiple Correct Options

(A)

x = - 1

(B)

x = 0

(C)

x = 2

(D)

x = 1

JEE Main 2021 (26 Aug Shift 2)

LEVELJEE Advanced

Let $[t]$ denote the greatest integer less than or equal to $t$ . Let $f (x) = x - [x]$ , $g (x) = 1 - x + [x]$ , and $h (x) = min {f (x), g (x)}$ , $x \in [- 2, 2]$ . Then $h$ is :

(A)

continuous in

[- 2, 2]

but not differentiable at more than four points in

(- 2, 2)

(B)

not continuous at exactly three points in

[- 2, 2]

(C)

continuous in

[- 2, 2]

but not differentiable at exactly three points in

(- 2, 2)

(D)

not continuous at exactly four points in

[- 2, 2]

JEE Main 2018 (16 April Shift 1)

LEVELJEE Advanced

If the function f defined as $f (x) = \frac{1}{x} - \frac{k - 1}{e ^{2 x} - 1}, x \neq = 0$ , is continuous at $x = 0$ , then the ordered pair $(k, f (0))$ is equal to :

(A)

(\frac{1}{3}, 2)

(B)

(3, 2)

(C)

(2, 1)

(D)

(3, 1)

JEE Advanced 2023

LEVELJEE Advanced

Let $f : (0, 1) \to R$ be the function defined as $f (x) = [4 x] (x - \frac{1}{4})^{2} (x - \frac{1}{2})$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Then which of the following statements is(are) true?

* Multiple Correct Options

(A)

The function

f

is discontinuous exactly at one point in

(0, 1)

(B)

There is exactly one point in

(0, 1)

at which the function

f

is continuous but NOT differentiable

(C)

The function

f

is NOT differentiable at more than three points in

(0, 1)

(D)

The minimum value of the function

f

- \frac{1}{512}

JEE Main 2018 (Paper 1)

LEVELJEE Advanced

Let $S = {t \in R : f (x) = ∣ x - π ∣ \cdot (e^{∣ x ∣} - 1) sin ∣ x ∣ is not differentiable at t}$ . Then the set $S$ is equal to :

(A)

{0, \pi}

(B)

ϕ (an empty set)

JEE Main 2021 (25 July Shift 2)

LEVELJEE Advanced

If $f (x) = {\int_{0}^{x} (5 + ∣1 - t ∣) d t, 5 x + 1, x > 2 x \leq 2$ , then

(A)

f (x)

is not continuous at

x = 2

(B)

f (x)

is everywhere differentiable

(C)

f (x)

is continuous but not differentiable at

x = 2

(D)

f (x)

is not differentiable at

x = 1

JEE Main 2021 (20 July Shift 1)

LEVELJEE Advanced

If the value of $lim_{x \to 0} (2 - cos x cos 2 x)^{\frac{x + 2}{x ^{2}}}$ is equal to $e^{a}$ , then $a$ is equal to

JEE Advanced 1983

LEVELJEE Advanced

Let $f (x) = {1 + x, 0 \leq x \leq 2 3 - x, 2 < x \leq 3$ . Determine the form of $g (x) = f (f (x))$ and hence find the points of discontinuity of $g$ , if any.

JEE Advanced 2020

LEVELJEE Advanced

Let $f : R \to R$ and $g : R \to R$ be functions satisfying $f (x + y) = f (x) + f (y) + f (x) f (y)$ and $f (x) = xg (x)$ for all $x, y \in R$ . If $lim_{x \to 0} g (x) = 1$ , then which of the following statements is/are TRUE?

* Multiple Correct Options

(A)

f

is differentiable at every

x \in R

(B)

(B) If

g (0) = 1

, then

g

is differentiable at every

x \in R

(C)

f^{'} (1)

is equal to 1

(D)

(D) The derivative

f^{'} (0)

is equal to 1

JEE Main 2021 (27 Aug Shift 1)

LEVELJEE Advanced

If $α, β$ are the distinct roots of $x^{2} + b x + c = 0$ , then $lim_{x \to β} \frac{e ^{2 (x^{2} + b x + c)} - 1 - 2 ( x ^{2} + b x + c )}{( x - β ) ^{2}}$ is equal to:

(A)

b^{2} + 4 c

(B)

2 (b^{2} + 4 c)

(C)

2 (b^{2} - 4 c)

(D)

b^{2} - 4 c

JEE Main 2021 (February)

LEVELJEE Advanced

The number of points, at which the function $f (x) = ∣2 x + 1∣ - 3∣ x + 2∣ + ∣ x^{2} + x - 2∣$ , $x \in R$ is not differentiable, is

JEE Advanced 1989

LEVELJEE Advanced

Draw a graph of the function $y = [x] + ∣1 - x ∣, - 1 \leq x \leq 3$ . Determine the points, if any, where this function is not differentiable.

JEE Advanced 2018

LEVELJEE Advanced

Let $f_{1} : R \to R$ , $f_{2} : (- \frac{π}{2}, \frac{π}{2}) \to R$ , $f_{3} : (- 1, e^{π /2} -2) \to R$ and $f_{4} : R \to R$ be functions defined by (i) $f_{1} (x) = sin 1 - e^{- x^{2}}$ (ii) $f_{2} (x) = {\frac{s i n x}{t a n ^{- 1} x} 1 if x \neq = 0 if x = 0$ , where the inverse trigonometric function $tan^{- 1} x$ assumes values in $(- \frac{π}{2}, \frac{π}{2})$ (iii) $f_{3} (x) = [sin (lo g_{e} (x + 2))]$ , where, for $t \in R$ , $[t]$ denotes the greatest integer less than or equal to $t$ , (iv) $f_{4} (x) = {x^{2} sin (\frac{1}{x}) 0 if x \neq = 0 if x = 0$ . Match the functions in List-I with the related properties in List-II.

List-I

(P)

The function

f_{1}

(Q)

The function

f_{2}

(R)

The function

f_{3}

(S)

The function

f_{4}

List-II

(1)

NOT continuous at

x = 0

(2)

continuous at

x = 0

and NOT differentiable at

x = 0

(3)

differentiable at

x = 0

and its derivative is NOT continuous at

x = 0

(4)

differentiable at

x = 0

and its derivative is continuous at

x = 0

JEE Advanced 1989

LEVELJEE Advanced

$A B C$ is an isosceles triangle inscribed in a circle of radius $r$ . If $A B = A C$ and $h$ is the altitude from $A$ to $B C$ then the triangle $A B C$ has perimeter $P = 2 (2 r h - h^{2} + 2 h r)$ and area $A = .........$ also $lim_{h \to 0} \frac{A}{P ^{3}} = .........$

JEE Main 2026 (24 January Shift 1)

LEVELJEE Advanced

If the function $f (x) = \frac{e ^{x} ( e ^{t a n x - x} - 1 ) + l o g _{e} ( s e c x + t a n x ) - x}{t a n x - x}$ is continuous at $x = 0$ , then the value of $f (0)$ is equal to

(A)

\frac{2}{3}

(B)

(C)

\frac{3}{2}

(D)

\frac{1}{2}

Micro-Topic Strategy Matrix

Evaluation of Limits & L'Hopital's Rule

Continuity at a Point and in an Interval

Differentiability of a Function

Relationship Between Continuity and Differentiability

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function f(x)=xcos(π(x+[x])) is discontinuous ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let [t] denote the greatest integer less than or equal to t. Let f(x)=x−[x], g(x)=1−x+[x], and h(x)=min{f(x),g(x)}, x∈[−2,2]. Then h is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the function f defined as f(x)=x1​−e2x−1k−1​,x=0, is continuous at x=0, then the ordered pair (k,f(0)) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={t∈R:f(x)=∣x−π∣⋅(e∣x∣−1)sin∣x∣ is not differentiable at t}. Then the set S is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)={∫0x​(5+∣1−t∣)dt,5x+1,​x>2x≤2​, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the value of limx→0​(2−cosxcos2x​)x2x+2​ is equal to ea, then a is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={1+x,0≤x≤23−x,2<x≤3​. Determine the form of g(x)=f(f(x)) and hence find the points of discontinuity of g, if any.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α,β are the distinct roots of x2+bx+c=0, then limx→β​(x−β)2e2(x2+bx+c)−1−2(x2+bx+c)​ is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of points, at which the function f(x)=∣2x+1∣−3∣x+2∣+∣x2+x−2∣, x∈R is not differentiable, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Draw a graph of the function y=[x]+∣1−x∣,−1≤x≤3. Determine the points, if any, where this function is not differentiable.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the function f(x)=tanx−xex(etanx−x−1)+loge​(secx+tanx)−x​ is continuous at x=0, then the value of f(0) is equal to

Let $[x]$ be the greatest integer less than or equals to $x$ . Then, at which of the following point(s) the function $f (x) = x cos (π (x + [x]))$ is discontinuous ?

Let $[t]$ denote the greatest integer less than or equal to $t$ . Let $f (x) = x - [x]$ , $g (x) = 1 - x + [x]$ , and $h (x) = min {f (x), g (x)}$ , $x \in [- 2, 2]$ . Then $h$ is :

If the function f defined as $f (x) = \frac{1}{x} - \frac{k - 1}{e ^{2 x} - 1}, x \neq = 0$ , is continuous at $x = 0$ , then the ordered pair $(k, f (0))$ is equal to :

Let $S = {t \in R : f (x) = ∣ x - π ∣ \cdot (e^{∣ x ∣} - 1) sin ∣ x ∣ is not differentiable at t}$ . Then the set $S$ is equal to :

If $f (x) = {\int_{0}^{x} (5 + ∣1 - t ∣) d t, 5 x + 1, x > 2 x \leq 2$ , then

If the value of $lim_{x \to 0} (2 - cos x cos 2 x)^{\frac{x + 2}{x ^{2}}}$ is equal to $e^{a}$ , then $a$ is equal to

Let $f (x) = {1 + x, 0 \leq x \leq 2 3 - x, 2 < x \leq 3$ . Determine the form of $g (x) = f (f (x))$ and hence find the points of discontinuity of $g$ , if any.

If $α, β$ are the distinct roots of $x^{2} + b x + c = 0$ , then $lim_{x \to β} \frac{e ^{2 (x^{2} + b x + c)} - 1 - 2 ( x ^{2} + b x + c )}{( x - β ) ^{2}}$ is equal to:

The number of points, at which the function $f (x) = ∣2 x + 1∣ - 3∣ x + 2∣ + ∣ x^{2} + x - 2∣$ , $x \in R$ is not differentiable, is

Draw a graph of the function $y = [x] + ∣1 - x ∣, - 1 \leq x \leq 3$ . Determine the points, if any, where this function is not differentiable.

If the function $f (x) = \frac{e ^{x} ( e ^{t a n x - x} - 1 ) + l o g _{e} ( s e c x + t a n x ) - x}{t a n x - x}$ is continuous at $x = 0$ , then the value of $f (0)$ is equal to