Limit, Continuity and Differentiability · Mathematics · NDA
MCQ (Single Correct Answer)
1
If $$f(x) = \sqrt {25 - {x^2}} $$, then what is $$\mathop {\lim }\limits_{x \to 1} {{f(x) - f(1)} \over {x - 1}}$$ equal to ?
NDA 2015 Paper 2
2
Consider the function
$$f(x) = \left\{ {\matrix{ {ax - 2,} & {for} & { - 2 < x < - 1} \cr { - 1,} & {for} & { - 1 \le x \le 1} \cr {a + 2{{(x - 1)}^2},} & {for} & {1 < x < 2} \cr } } \right.$$
What is the value of a for which f(x) is continuous at x = $$-$$1 and x = 1?
$$f(x) = \left\{ {\matrix{ {ax - 2,} & {for} & { - 2 < x < - 1} \cr { - 1,} & {for} & { - 1 \le x \le 1} \cr {a + 2{{(x - 1)}^2},} & {for} & {1 < x < 2} \cr } } \right.$$
What is the value of a for which f(x) is continuous at x = $$-$$1 and x = 1?
NDA 2015 Paper 2
3
The function $$f(x) = {{1 - \sin x + \cos x} \over {1 + \sin x + \cos x}}$$ is not defined at x = $$\pi$$. The value of f($$\pi$$), so that f(x) is continuous at x = $$\pi$$, is
NDA 2015 Paper 2
4
Consider the following functions
1. $$f(x) = \left\{ \matrix{ {1 \over x},\,if\,x \ne 0 \hfill \cr 0,\,if\,x = 0 \hfill \cr} \right.$$
$$f(x) = \left\{ {\matrix{ {2x + 5,} & {if\,x > 0} \cr {{x^2} + 2x + 5,} & {if\,x \le 0} \cr } } \right.$$
Which of the above functions is/are derivable at x = 0 ?
1. $$f(x) = \left\{ \matrix{ {1 \over x},\,if\,x \ne 0 \hfill \cr 0,\,if\,x = 0 \hfill \cr} \right.$$
$$f(x) = \left\{ {\matrix{ {2x + 5,} & {if\,x > 0} \cr {{x^2} + 2x + 5,} & {if\,x \le 0} \cr } } \right.$$
Which of the above functions is/are derivable at x = 0 ?
NDA 2015 Paper 2
5
If $$f(x) = {{\sin ({e^{x - 2}} - 1)} \over {\ln (x - 1)}}$$, then $$\mathop {\lim }\limits_{x \to 2} $$ f(x) is equal to
NDA 2015 Paper 2
6
The value of A is
NDA 2015 Paper 2
7
The value of B is
NDA 2015 Paper 2
8
Consider the following in respect of the function
$$f(x) = \left\{ \matrix{ 2 + x,x \ge 0 \hfill \cr 2 - x,x < 0 \hfill \cr} \right.$$.
I. $$\mathop {\lim }\limits_{x \to 1} f(x)$$ does not exist.
II. f(x) is differentiable at x = 0.
III. f(x) is continuous at x = 0.
Which of the above statements is/are correct?
$$f(x) = \left\{ \matrix{ 2 + x,x \ge 0 \hfill \cr 2 - x,x < 0 \hfill \cr} \right.$$.
I. $$\mathop {\lim }\limits_{x \to 1} f(x)$$ does not exist.
II. f(x) is differentiable at x = 0.
III. f(x) is continuous at x = 0.
Which of the above statements is/are correct?
NDA 2016 Paper 2
9
Let f : A $$\to$$ R, where A = R \ {0} is such that $$f(x) = {{x + |x|} \over x}$$. On which one of the following sets is f(x) continuous?
NDA 2016 Paper 2
10
Which of the following statements is correct?
NDA 2016 Paper 2
11
Which one of the following statements is correct?
NDA 2016 Paper 2
12
Which of the following statements are correct?
I. (fof) (x) = f(x).
II. (gog) (x) = g(x) only when x = 0.
III. (go(fog)) (x) can taken only three values.
Select the correct answer using the code given below.
I. (fof) (x) = f(x).
II. (gog) (x) = g(x) only when x = 0.
III. (go(fog)) (x) can taken only three values.
Select the correct answer using the code given below.
NDA 2016 Paper 2
13
Which of the following statements is/are correct?
I. g(x) is differentiable at x = 0
II. g(x) is differentiable at x = 2.
Select the correct answer using the code given below.
I. g(x) is differentiable at x = 0
II. g(x) is differentiable at x = 2.
Select the correct answer using the code given below.
NDA 2016 Paper 2
14
What is the value of the differential coefficient of g(x) at x = $$-$$2 ?
NDA 2016 Paper 2
15
Which of the following statements are correct?
I. g(x) is continuous at x = 0.
II. g(x) is continuous at x = 2.
III. g(x) is continuous at x = $$-$$1.
Select the correct answer using the code given below.
I. g(x) is continuous at x = 0.
II. g(x) is continuous at x = 2.
III. g(x) is continuous at x = $$-$$1.
Select the correct answer using the code given below.
NDA 2016 Paper 2
16
Let $$f(x) = \left\{ {\matrix{
{{{{e^x} - 1} \over x},} & {x > 0} \cr
{0,} & {x = 0} \cr
} } \right.$$, be a real valued function, then which of the following statements is/are correct?
I. f(x) is right continuous at x = 0
II. f(x) is discontinuous at x = 1
Select the correct answer using the code given below.
I. f(x) is right continuous at x = 0
II. f(x) is discontinuous at x = 1
Select the correct answer using the code given below.
NDA 2016 Paper 2
17
Which of the following statements are not correct?
1. y as a function of x is not defined for all real x.
2. y as a function of x is not continuous at x = 0.
3. y as a function of x is differentiable for all x.
Select the correct answer using the codes given below.
1. y as a function of x is not defined for all real x.
2. y as a function of x is not continuous at x = 0.
3. y as a function of x is differentiable for all x.
Select the correct answer using the codes given below.
NDA 2016 Paper 1
18
What is the derivative of y as a function of x with respect to x for x < 0 ?
NDA 2016 Paper 1
19
What is $$\mathop {\lim }\limits_{x \to {0^ + }} f(x)$$ equal to
NDA 2016 Paper 1
20
What is $$\mathop {\lim }\limits_{x \to {0^ - }} f(x)$$ equal to?
NDA 2016 Paper 1
21
Consider the following statements
1. The function f(x) is continuous at x = 0.
2. The function f(x) is continuous at $$x = {\pi \over 2}$$
Which of the above statements is/are correct?
1. The function f(x) is continuous at x = 0.
2. The function f(x) is continuous at $$x = {\pi \over 2}$$
Which of the above statements is/are correct?
NDA 2016 Paper 1
22
Consider the following statements
1. The function f(x) is differentiable at x = 0.
2. The function f(x) is differentiable at x = $${\pi \over 2}$$.
Which of the above statements is/are correct?
1. The function f(x) is differentiable at x = 0.
2. The function f(x) is differentiable at x = $${\pi \over 2}$$.
Which of the above statements is/are correct?
NDA 2016 Paper 1
23
If $$\mathop {\lim }\limits_{x \to 0} \phi (x) = {a^2}$$, where a $$\ne$$ 0, then what is $$\mathop {\lim }\limits_{x \to 0} \phi \left( {{x \over a}} \right)$$ equal to ?
NDA 2016 Paper 1
24
What is $$\mathop {\lim }\limits_{x \to 0} {e^{ - {1 \over {{x^2}}}}}$$ equal to ?
NDA 2016 Paper 1
25
Consider the function $$f(x) = \left| {x - 1} \right| + {x^2}$$ where x $$\in$$ R, then which one of the following statements is correct?
NDA 2016 Paper 1
26
A function is defined as follows
$$f(x) = \left\{ {\matrix{ { - {x \over {\sqrt {{x^2}} }},} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$$
Which one of the following is correct in respect of the above function?
$$f(x) = \left\{ {\matrix{ { - {x \over {\sqrt {{x^2}} }},} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$$
Which one of the following is correct in respect of the above function?
NDA 2017 Paper 2
27
Consider the following statements
1. $$x + {x^2}$$ is continuous at x = 0
2. $$x + \cos {1 \over x}$$ is discontinuous at x = 0
3. $${x^2} + \cos {1 \over x}$$ is continuous at x = 0
Which of the above are correct?
1. $$x + {x^2}$$ is continuous at x = 0
2. $$x + \cos {1 \over x}$$ is discontinuous at x = 0
3. $${x^2} + \cos {1 \over x}$$ is continuous at x = 0
Which of the above are correct?
NDA 2017 Paper 2
28
A function is defined in (0, $$\infty$$) by
$$f(x) = \left( {\matrix{ {1 - {x^2}} & {for} & {0 < x \le 1} \cr {\ln x} & {for} & {1 < x \le 2} \cr {\ln 2 - 1 + 0.5x} & {for} & {2 < x < \infty } \cr } } \right.$$
Which one of the following is correct in respect of the derivative of the function, i.e., f'(x) ?
$$f(x) = \left( {\matrix{ {1 - {x^2}} & {for} & {0 < x \le 1} \cr {\ln x} & {for} & {1 < x \le 2} \cr {\ln 2 - 1 + 0.5x} & {for} & {2 < x < \infty } \cr } } \right.$$
Which one of the following is correct in respect of the derivative of the function, i.e., f'(x) ?
NDA 2017 Paper 2
29
If $$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\sin x} \over x} = l$$ and $$\mathop {\lim }\limits_{x \to \infty } {{\cos x} \over x} = m$$, then which one of the following is correct?
NDA 2017 Paper 2
30
The left-hand derivative of f(x) = [x] sin ($$\pi$$x) at x = k, where k is an integer and [x] is the greatest integer function, is
NDA 2017 Paper 2
31
The set of all points, where the function $$f(x) = \sqrt {1 - {e^{ - {x^2}}}} $$ is differentiable, is
NDA 2017 Paper 2
32
If $$f(x) = x(\sqrt x - \sqrt {x + 1} )$$, then f(x) is
NDA 2017 Paper 2
33
Let g be the greatest integer function. Then, the function f(x) = (g(x))2 $$-$$ g(x) is discontinuous at
NDA 2017 Paper 2
34
What is $$\mathop {\lim }\limits_{x \to 0} {{{e^x} - (1 + x)} \over {{x^2}}}$$ equal to ?
NDA 2017 Paper 1
35
If $$F(x) = \sqrt {9 - {x^2}} $$, then what is $$\mathop {\lim }\limits_{x \to 1} {{F(x) - F(1)} \over {x - 1}}$$ equal to ?
NDA 2017 Paper 1
36
Let f(x + y) = f(x) f(y) for all x and y. Then, what is f'(5) equal to [where f' (x) is the derivative of f(x)]?
NDA 2017 Paper 1
37
Let f(x) be defined as follows
$$f(x) = \left\{ {\matrix{ {2x + 1,} & { - 3 < x < - 2} \cr {x - 1,} & { - 2 \le x < 0} \cr {x + 2,} & {0 \le x < 1} \cr } } \right.$$
Which one of the following statements is correct in respect of the above function?
$$f(x) = \left\{ {\matrix{ {2x + 1,} & { - 3 < x < - 2} \cr {x - 1,} & { - 2 \le x < 0} \cr {x + 2,} & {0 \le x < 1} \cr } } \right.$$
Which one of the following statements is correct in respect of the above function?
NDA 2017 Paper 1
38
Consider the following statements
1. If $$\mathop {\lim }\limits_{x \to a} f(x)$$ and $$\mathop {\lim }\limits_{x \to a} g(x)$$ both exist, then $$\mathop {\lim }\limits_{x \to a} \{ f(x)g(x)\} $$ exists.
2. If $$\mathop {\lim }\limits_{x \to a} \{ f(x)g(x)\} $$ exists, then both $$\mathop {\lim }\limits_{x \to a} f(x)$$ and $$\mathop {\lim }\limits_{x \to a} g(x)$$ must exist.
Which of the above statement is/are correct?
1. If $$\mathop {\lim }\limits_{x \to a} f(x)$$ and $$\mathop {\lim }\limits_{x \to a} g(x)$$ both exist, then $$\mathop {\lim }\limits_{x \to a} \{ f(x)g(x)\} $$ exists.
2. If $$\mathop {\lim }\limits_{x \to a} \{ f(x)g(x)\} $$ exists, then both $$\mathop {\lim }\limits_{x \to a} f(x)$$ and $$\mathop {\lim }\limits_{x \to a} g(x)$$ must exist.
Which of the above statement is/are correct?
NDA 2017 Paper 1
39
Consider the function $$f(x) = \left\{ {\matrix{
{{{\sin 2x} \over {5x}},} & {if\,x \ne 0} \cr
{{2 \over {15}},} & {if\,x = 0} \cr
} } \right.$$
Which one of the following is correct in respect of the function?
Which one of the following is correct in respect of the function?
NDA 2018 Paper 2
40
For the function f(x) = | x $$-$$ 3 |, which one of the following is not correct?
NDA 2018 Paper 2
41
If the function $$f(x) = {{2x - {{\sin }^{ - 1}}x} \over {2x + {{\tan }^{ - 1}}x}}$$ is continuous at each point in its domain then what is the value of f(0) ?
NDA 2018 Paper 2
42
If $$f(x) = \sqrt {25 - {x^2}} $$, then what is $$\mathop {\lim }\limits_{x \to 1} {{f(x) - f(1)} \over {x - 1}}$$ equal to ?
NDA 2018 Paper 2
43
What is $$\mathop {\lim }\limits_{\theta \to 0} {{\sqrt {1 - \cos \theta } } \over \theta }$$ equal to ?
NDA 2018 Paper 2
44
Let f(x + y) = f(x) f(y) and f(x) = 1 + xg(x) $$\phi$$ (x), where $$\mathop {\lim }\limits_{x \to 0} $$ g(x) = a and $$\mathop {\lim }\limits_{x \to 0} $$ $$\phi$$ (x) = b. Then, what is f'(x) equal to ?
NDA 2018 Paper 2
45
What is $$\mathop {\lim }\limits_{x \to {\pi \over 6}} {{2{{\sin }^2}x + \sin x - 1} \over {2{{\sin }^2}x - 3\sin x + 1}}$$ equal to ?
NDA 2018 Paper 2
46
Consider the function $$f(x) = \left\{ {\matrix{
{{x^2}In|x|} & {x \ne 0} \cr
0 & {x = 0} \cr
} } \right.$$ What is f' (0) equal to ?
NDA 2018 Paper 1
47
If $$f(x) = {{{x^2} - 9} \over {{x^2} - 2x - 3}},x \ne 3$$ is continuous at x = 3, then which one of the following is correct?
NDA 2018 Paper 1
48
What is $$\mathop {\lim }\limits_{x \to 0} {{\tan x} \over {\sin 2x}}$$ equal to
NDA 2018 Paper 1
49
What is $$\mathop {\lim }\limits_{h \to 0} {{\sqrt {2x + 3h} - \sqrt {2x} } \over {2h}}$$ equal to ?
NDA 2018 Paper 1
50
For what value of k is the function $$f(x) = \left\{ \matrix{
2x + {1 \over 4},x < 0 \hfill \cr
k,x = 0 \hfill \cr
{\left( {x + {1 \over 2}} \right)^2},x > 0 \hfill \cr} \right.$$ continuous?
NDA 2019 Paper 2
51
What is the value of $$\mathop {\lim }\limits_{x \to 0} {{\sin x^\circ } \over {\tan 3x^\circ }}$$ ?
NDA 2019 Paper 2
52
Consider the following statements in respect of the function $$f(x) = \sin \left( {{1 \over x}} \right)$$ for x $$\ne$$ 0 and f(0) = 0 :
1. $$\mathop {\lim }\limits_{x \to 0} f(x)$$ exits
2. f(x) is continuous at x = 0
hich of the above statement is/are correct?
1. $$\mathop {\lim }\limits_{x \to 0} f(x)$$ exits
2. f(x) is continuous at x = 0
hich of the above statement is/are correct?
NDA 2019 Paper 2
53
$$\mathop {\lim }\limits_{x \to 0} {{1 - {{\cos }^3}4x} \over {{x^2}}}$$ is equal to
NDA 2019 Paper 1
54
The value of k which makes $$f(x) = \left\{ {\matrix{
{\sin x,} & {x \ne 0} \cr
{k,} & {x = 0} \cr
} } \right.$$ continuous at x = 0, is
NDA 2019 Paper 1
55
Consider the following statements :
1. $f(x)$ is increasing in the interval $(e, \infty)$
2. $f(x)$ is decreasing in the interval $(1, e)$
3. $9 \ln 7 > 7 \ln 9$
Which of the statements given above are correct ?
NDA Mathematics 21 April 2024
56
What is $\lim\limits_{x \to 0-} h(x) + \lim\limits_{x \to 0+} h(x)$ equal to?
NDA Mathematics 21 April 2024
57
What is f(0.5) equal to ?
NDA Mathematics 3 September 2023
58
What is $\displaystyle \lim _{x \rightarrow 1} \frac{f(x)-1}{g(x)}$ equal to?
NDA Mathematics 3 September 2023
59
What is $\displaystyle \lim _{x \rightarrow 1} f(x)^{\frac{1}{g(x)}}$ equal to?
NDA Mathematics 3 September 2023
60
What is $\displaystyle \lim _{x \rightarrow 0+}$ h(x) equal to ?
NDA Mathematics 3 September 2023
61
What is $\displaystyle \lim _{x \rightarrow 0-}$ h(x) equal to ?
NDA Mathematics 3 September 2023
62
What is the value of a ?
NDA Mathematics 3 September 2023
63
What is the value of b?
NDA Mathematics 3 September 2023
64
If the function f(x) is differentiable at x = 1, then what is the value of (a + b)?
NDA Mathematics 3 September 2023
65
What is $\displaystyle \lim _{x \rightarrow 0} $ f(x) equal to ?
NDA Mathematics 3 September 2023
66
What is $\lim _{x \rightarrow 0} \frac{f(x)}{x}$ equal to ?
NDA Mathematics 16 April 2023
67
What is $\lim _{x \rightarrow 0} \frac{f(x)}{x^2}$ equal to ?
NDA Mathematics 16 April 2023
68
What is the value of p ?
NDA Mathematics 16 April 2023
69
What is the value of q ?
NDA Mathematics 16 April 2023
70
What is $\lim\limits_{x \rightarrow 5} \frac{5-x}{|x-5|}$ equal to ?
NDA Mathematics 16 April 2023
71
What is $\lim\limits_{x \rightarrow 1} \frac{x^9-1}{x^3-1}$ equal to ?
NDA Mathematics 16 April 2023
72
What is $\displaystyle\lim_{x \rightarrow 0} \frac{x}{\sqrt{1−\cos 4x}}$ equal to ?
NDA Mathematics 4 September 2022
73
What is $\displaystyle\lim_{x\rightarrow \frac{\pi}{2}} \frac{4x−2\pi}{\cos x}$ equal to ?
NDA Mathematics 4 September 2022
74
If f(x) = $\frac{x^2+x+|x|}{x}$, then what is $\displaystyle\lim_{x \rightarrow 0}$ f(x) equal to ?
NDA Mathematics 4 September 2022
75
What is $\displaystyle\lim_{h \rightarrow 0} \frac{\sin^2(x+h)−\sin^2x}{h}$ equal to ?
NDA Mathematics 4 September 2022
76
Let y = [x + 1], -4 < x < -3 where [.] is the greatest integer function. What is the derivative of y with respect to x at x = -3.5?
NDA Mathematics 10 April 2022
77
What is $\lim\limits_{x \to 0}x^3(\text{cosec} \ x)^2$ equal to ?
NDA Mathematics 10 April 2022
78
What is $\lim\limits_{x \to 1}\frac{x^3 - 1}{\sqrt x - 1}$ equal to?
NDA Mathematics 10 April 2022
79
If a differentiable function f(x) satisfies $\mathop {\lim }\limits_{x \to - 1} \dfrac{f(x)+1}{x^2-1}=-\dfrac{3}{2}$ then what is $\mathop {\lim }\limits_{x \to - 1} f(x)$ equal to?
NDA Mathematics 18 April 2021
80
If the function $\rm f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {a + bx,\;\;}&{x < 1}\\ {5,}&{x = 1}\\ {b - ax,}&{x > 1} \end{array}} \right.$ is continuous, then what is the value of (a + b)?
NDA Mathematics 18 April 2021
81
If $\rm\mathop {\lim }\limits_{x \to a} \frac{a^x -x^a}{x^x -a^a}= - 1$, then what is the value of a?
NDA Mathematics 18 April 2021
82
What is $\rm\mathop {\lim }\limits_{x \to 2}\frac{x^3 + x^2}{x^2 + 3x + 2}$ equal to?
NDA Mathematics 18 April 2021
83
If f(x) = e|x|, then which one of the following is correct?
NDA Mathematics 18 April 2021