Let f(x) be defined as follows: \({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {2{\rm{x}} + 1,{\rm{\;\;}} - 3 < {\rm{x}} < - 2}\\ {{\rm{x}} - 1,{\rm{\;\;}} - 2 \le {\rm{x}} < 0}\\ {{\rm{x}} + 2,{\rm{\;\;\;}}0 \le {\rm{x}} < 1} \end{array}} \right.\) Which one of the following statements is correct in respect of the above function?

It is discontinuous at every point

It is discontinuous at x = -2 but continuous at every other point.

It is continuous only in the interval (-3, -2)

It is discontinuous at x = 0 but continuous at every other point

It is discontinuous at every point

Let f: R → R be defined by \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {x\sin \left( {\frac{1}{x}} \right)}&{if\;x > 0}\\ 0&{x \le 0} \end{array}} \right.\) Then

f is not continuous but differentiable at x = 0

f is neither continuous nor differentiable at x = 0

f is continuous nor differentiable at x = 0

f is continuous but not differentiable at x = 0

f is not continuous but differentiable at x = 0

\(\frac{1}{{{{\log }_2}x}} + \frac{1}{{{{\log }_3}x}} + \frac{1}{{{{\log }_4}x}} + \ldots .. + \frac{1}{{{{\log }_{50}}x}},x \ne 1\) is equal to

\(\frac{{1}}{{{{\log }_{49!}}x}}\)

\(\frac{{50}}{{{{\log }_{50}}x}}\)

\(\frac{{49}}{{{{\log }_{49}}x}}\)

\(\frac{{1}}{{{{\log }_{50!}}x}}\)

\(\frac{{1}}{{{{\log }_{49!}}x}}\)

If \({\rm{G}}\left( {\rm{x}} \right) = \sqrt {(25 - {{\rm{x}}^2}} \) then what is \(\mathop {\lim }\limits_{{\rm{x}} \to 1} \frac{{{\rm{G}}\left( {\rm{x}} \right) - {\rm{G}}\left( 1 \right)}}{{{\rm{x}} - 1}}{\rm{\;}}\)equal to?

\(- \frac{1}{{2\sqrt 6 {\rm{\;\;}}}}\)

\(\displaystyle\lim_{x\rightarrow 0} \dfrac{a^x-b^x}{e^x-1}\) is equal to :

\(\log \left(\dfrac{b}{a}\right)\)

\(\log \left(\dfrac{a}{b}\right)\)

\(\log \left(\dfrac{b}{a}\right)\)

If \(\mathop {\lim }\limits_{{\rm{x}} \to \frac{{\rm{\pi }}}{2}} \frac{{\sin {\rm{x}}}}{{\rm{x}}} = {\rm{l}}\) and \(\mathop {\lim }\limits_{{\rm{x}} \to \infty } \frac{{\cos {\rm{x}}}}{{\rm{x}}} = {\rm{m}}\), then which one of the following is correct?

\({\rm{l}} = \frac{2}{{\rm{\pi }}},{\rm{\;m}} = \infty \)

\({\rm{l}} = \frac{2}{{\rm{\pi }}},{\rm{\;m}} = 0\)

A function is defined as follows:\({\rm{f}}\left( {\rm{x}} \right):\left\{ {\begin{array}{*{20}{c}} { - \frac{{\rm{x}}}{{\sqrt {{{\rm{x}}^2}} }},{\rm{\;x}} \ne 0}\\ {0.{\rm{\;x}} = 0} \end{array}} \right.\)Which one of the following is correct in respect of the above function?

f(x) is continuous at x = 0 but not differentiable at x = 0

f(x) is continuous as well as differentiable at x = 0

At x = 0, the function \(f(x) =\left | \frac{\sin2\pi x}{L}\right |\) (- ∞ 0) is

Not continuous and not differentiable

Not continuous but differentiable.

Continuous but not differentiable.

If a function f(x) = \(\begin{Bmatrix} \rm 3x + 2, x \geq1\\ \rm 5, \;\;\;\;\;\;\;\;x

Continuous but f(1) cannot be determined

41 + Mcqs in Limit And Continuity in Mathematics Page 1 McqOptions

1.	Consider the function f defined by \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {{x^2} - 1,}&{x < 3}\\ {2ax,}&{x \ge 3} \end{array}} \right.\) for all real numbers x. If f is continuous at x = 3, then value of a
A.	8
B.	3 / 4
C.	1 / 8
D.	4 / 3
Answer» E.

Discussion

2.	Let f(x) be defined as follows: \({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {2{\rm{x}} + 1,{\rm{\;\;}} - 3 < {\rm{x}} < - 2}\\ {{\rm{x}} - 1,{\rm{\;\;}} - 2 \le {\rm{x}} < 0}\\ {{\rm{x}} + 2,{\rm{\;\;\;}}0 \le {\rm{x}} < 1} \end{array}} \right.\) Which one of the following statements is correct in respect of the above function?
A.	It is discontinuous at x = -2 but continuous at every other point.
B.	It is continuous only in the interval (-3, -2)
C.	It is discontinuous at x = 0 but continuous at every other point
D.	It is discontinuous at every point
Answer» D. It is discontinuous at every point

Discussion

3.	Let f: R → R be defined by \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {x\sin \left( {\frac{1}{x}} \right)}&{if\;x > 0}\\ 0&{x \le 0} \end{array}} \right.\) Then
A.	f is neither continuous nor differentiable at x = 0
B.	f is continuous nor differentiable at x = 0
C.	f is continuous but not differentiable at x = 0
D.	f is not continuous but differentiable at x = 0
Answer» D. f is not continuous but differentiable at x = 0

Discussion

4.	If \({\rm{F}}\left( {\rm{x}} \right) = \sqrt {9 - {{\rm{x}}^2}} \), then what is \(\mathop {\lim }\limits_{{\rm{x}} \to 1} \frac{{{\rm{F}}\left( {\rm{x}} \right) - {\rm{F}}\left( 1 \right)}}{{{\rm{x}} - 1}}\) equal to?
A.	\( - \frac{1}{{4\sqrt 2 }}\)
B.	\(\frac{1}{8}\)
C.	\(- \frac{1}{{2\sqrt 2 }}\)
D.	\(\frac{1}{{2\sqrt 2 }}\)
Answer» D. \(\frac{1}{{2\sqrt 2 }}\)

Discussion

5.	\(\frac{1}{{{{\log }_2}x}} + \frac{1}{{{{\log }_3}x}} + \frac{1}{{{{\log }_4}x}} + \ldots .. + \frac{1}{{{{\log }_{50}}x}},x \ne 1\) is equal to
A.	\(\frac{{50}}{{{{\log }_{50}}x}}\)
B.	\(\frac{{49}}{{{{\log }_{49}}x}}\)
C.	\(\frac{{1}}{{{{\log }_{50!}}x}}\)
D.	\(\frac{{1}}{{{{\log }_{49!}}x}}\)
Answer» D. \(\frac{{1}}{{{{\log }_{49!}}x}}\)

Discussion

6.	If \(\rm \displaystyle\lim_{x \rightarrow 1} \dfrac{x^4-1}{x-1} = \lim_{x\rightarrow k} \dfrac{x^3-k^3}{x^2-k^2}\), where k ≠ 0, then what is the value of k?
A.	\(\dfrac{2}{3}\)
B.	\(\dfrac{4}{3}\)
C.	\(\dfrac{8}{3}\)
D.	4
Answer» D. 4

Discussion

7.	If \({\rm{G}}\left( {\rm{x}} \right) = \sqrt {(25 - {{\rm{x}}^2}} \) then what is \(\mathop {\lim }\limits_{{\rm{x}} \to 1} \frac{{{\rm{G}}\left( {\rm{x}} \right) - {\rm{G}}\left( 1 \right)}}{{{\rm{x}} - 1}}{\rm{\;}}\)equal to?
A.	\(- \frac{1}{{2\sqrt 6 {\rm{\;\;}}}}\)
B.	\(\frac{1}{5}\)
C.	\(- \frac{1}{{\sqrt 6 }}\)
D.	\(\frac{1}{{\sqrt 6 }}\)
Answer» B. \(\frac{1}{5}\)

Discussion

8.	\(\displaystyle\lim_{x\rightarrow 0} \dfrac{a^x-b^x}{e^x-1}\) is equal to :
A.	\(\log \left(\dfrac{a}{b}\right)\)
B.	\(\log \left(\dfrac{b}{a}\right)\)
C.	log (a, b)
D.	log (a + b)
Answer» B. \(\log \left(\dfrac{b}{a}\right)\)

Discussion

9.	If \({\rm{f}}\left( {\rm{x}} \right) = \frac{{{\rm{sin}}\left( {{{\rm{e}}^{{\rm{x}} - 2}} - 1} \right)}}{{{\rm{In}}\left( {{\rm{x}} - 1} \right)}}\), then \(\mathop {\lim }\limits_{{\rm{x}} \to 2} {\rm{f}}\left( {\rm{x}} \right)\) is equal to
A.	-2
B.	-1
C.	0
D.	1
Answer» E.

Discussion

10.	\(\mathop {{\rm{lim}}}\limits_{x \to \infty } \left( {\frac{{2 + {x^2}}}{{1 + x\;}} - Ax - B} \right) = 3\)What is the value of B?
A.	-4
B.	1
C.	2
D.	3
Answer» B. 1

Discussion

11.	\(\mathop {{\rm{lim}}}\limits_{x \to \infty } \left( {\frac{{2 + {x^2}}}{{1 + x\;}} - Ax - B} \right) = 3\)What is the value of A?
A.	-1
B.	1
C.	2
D.	3
Answer» C. 2

Discussion

12.	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?
A.	\(-\dfrac{3}{2}\)
B.	-1
C.	0
D.	1
Answer» C. 0

Discussion

13.	If \(\mathop {\lim }\limits_{{\rm{x}} \to \frac{{\rm{\pi }}}{2}} \frac{{\sin {\rm{x}}}}{{\rm{x}}} = {\rm{l}}\) and \(\mathop {\lim }\limits_{{\rm{x}} \to \infty } \frac{{\cos {\rm{x}}}}{{\rm{x}}} = {\rm{m}}\), then which one of the following is correct?
A.	l = 1, m = 1
B.	\({\rm{l}} = \frac{2}{{\rm{\pi }}},{\rm{\;m}} = \infty \)
C.	\({\rm{l}} = \frac{2}{{\rm{\pi }}},{\rm{\;m}} = 0\)
D.	l = 1, m = ∞
Answer» D. l = 1, m = ∞

Discussion

14.	If \({\rm{f}}\left( {\rm{x}} \right) = \sqrt {25 - {{\rm{x}}^2}} {\rm{\;}},{\rm{\;}}\) then what is \(\mathop {{\rm{Lim}}}\limits_{{\rm{x}} \to 1} \frac{{{\rm{f}}\left( {\rm{x}} \right) - {\rm{f}}\left( 1 \right)}}{{{\rm{x}} - 1}}\) equal to?
A.	\(\frac{1}{5}\)
B.	\(\frac{1}{24}\)
C.	\(\sqrt {24} \)
D.	\(- \frac{1}{{\sqrt {24} }}\)
Answer» E.

Discussion

15.	If \(\mathop {\lim }\limits_{{\rm{x}} \to 0} \phi \left( {\rm{x}} \right) = {{\rm{a}}^2}\), where a ≠ 0, then what is \(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 0} \phi \left( {\frac{{\rm{x}}}{{\rm{a}}}} \right)\) Equal to
A.	a2
B.	a-2
C.	–a2
D.	-a
Answer» B. a-2

Discussion

16.	If f(x) =\(\left\lbrace \begin{matrix}\dfrac{\sin [\rm x]}{[\rm x]}, \ \ [\rm x] \neq 0 \\\ 0, \ \ [\rm x] = 0\end{matrix} \right.\), where [x] is the largest integer but not larger than x, then \(\rm \displaystyle\lim_{x \rightarrow 0} f(x)\) is
A.	-1
B.	0
C.	1
D.	Does not exist
Answer» E.

Discussion

17.	If \(x + \frac{1}{x} = \sqrt{3}\), then the value of x18 + x12 + x6 + 1 is
A.	0
B.	1
C.	2
D.	3
Answer» B. 1

Discussion

18.	f(x) = x + \|x\| is continuous for
A.	x ∈ (-∞, ∞)
B.	x ∈ (-∞, ∞) - {0}
C.	only x > 0
D.	No value of x
Answer» B. x ∈ (-∞, ∞) - {0}

Discussion

19.	Consider the following statements for f(x) = e-\|x\| ;1. The function is continuous at x = 0.2. The function is differentiable at x = 0.Which of the above statements is / are correct?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» B. 2 only

Discussion

20.	Let f : A → R, where A = R\(0) is such that \({\rm{f}}\left( {\rm{x}} \right) = \frac{{{\rm{x}} + \left\| {\rm{x}} \right\|}}{{\rm{x}}}\). On which one of the following sets is f(x) continuous?
A.	A
B.	B = {x ∈ R : x ≥ 0}
C.	C = {x ∈ R : x ≤ 0}
D.	D = R
Answer» B. B = {x ∈ R : x ≥ 0}

Discussion

21.	If \(\displaystyle\lim_{x\rightarrow \infty}\left(1+ \dfrac{a}{x}+\dfrac{b}{x^2}\right)^{2x}=e^2\), then the value of a and b are
A.	a ∈ R, b = 2
B.	a = 1, b ∈ R
C.	a ∈ R, b ∈ R
D.	None of these
Answer» C. a ∈ R, b ∈ R

Discussion

22.	If a + b + c = 5 and ab + bc + ca = 10, then the value of a3 + b3 + c3 - 3abc is
A.	-25
B.	25
C.	0
D.	75
Answer» B. 25

Discussion

23.	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)?
A.	5
B.	10
C.	15
D.	20
Answer» B. 10

Discussion

24.	If \(\rm f(x) = \left \{ \begin{matrix} \rm x^2; & \rm x \leq 0 \\ \rm 2\sin x; & \rm x > 0 \end{matrix}\right.\), then x = 0 is a point of:
A.	Minima.
B.	Maxima.
C.	Discontinuity.
D.	None of these.
Answer» B. Maxima.

Discussion

25.	If \(\rm \lim_{x \to a} \frac{a^x -x^a}{x^x -a^a}= - 1\), then what is the value of a?
A.	-1
B.	0
C.	1
D.	2
Answer» D. 2

Discussion

26.	If \(\rm f(x) = \left\{ \begin{matrix} \rm \dfrac{x-x^2}{2x}; & \rm x \ne 0 \\ \rm K; & \rm x = 0 \end{matrix}\right.\) is a continuous function at x = 0, then the value of k is:
A.	2
B.	\(\dfrac12\)
C.	1
D.	None of these
Answer» C. 1

Discussion

27.	\(\lim_{x \rightarrow 3} \frac {\sqrt {3x} - 3}{\sqrt {2x - 4} - \sqrt 2}\) is equal to
A.	\(\sqrt 3\)
B.	\(\frac {\sqrt 3} 2\)
C.	\(\frac 1 {2\sqrt 2}\)
D.	\(\frac 1 {\sqrt 2}\)
Answer» E.

Discussion

28.	Consider the following statements:1. f(x) = [x], where [.] is the greatest integer function, is discontinuous at x = n, where n ϵ Z.2. f(x) = cot x is discontinuous at x = nπ, where n ϵ Z. Which of the above statements is/are correct?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» D. Neither 1 nor 2

Discussion

29.	Find the value of alogaN
A.	N
B.	a
C.	1
D.	0
Answer» B. a

Discussion

30.	\(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {\frac{1}{n}} {e^{r/n}}\) is
A.	e
B.	e - 1
C.	1 - e
D.	e + 1
Answer» C. 1 - e

Discussion

31.	If \(x + \frac{1}{x} = 2\), then the value of \({x^3} + \frac{1}{{{x^3}}}\) is
A.	64
B.	16
C.	8
D.	2
Answer» E.

Discussion

32.	Let \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\cos \left[ x \right],}&{x \ge 0}\\ {\left\| x \right\| + a,}&{x < 0} \end{array}} \right.,\) where [x] denotes the greatest integer ≤ x. If f should be continuous at x = 0, then a must be
A.	0
B.	1
C.	2
D.	-1
Answer» C. 2

Discussion

33.	If \(\rm f(x)=\dfrac{\sin x}{x}\), where x ∈ R, is to be continuous at x = 0, then the value of the function at x = 0
A.	should be 0
B.	should be 1
C.	should be 2
D.	cannot be determined
Answer» C. should be 2

Discussion

34.	A function is defined as follows:\({\rm{f}}\left( {\rm{x}} \right):\left\{ {\begin{array}{*{20}{c}} { - \frac{{\rm{x}}}{{\sqrt {{{\rm{x}}^2}} }},{\rm{\;x}} \ne 0}\\ {0.{\rm{\;x}} = 0} \end{array}} \right.\)Which one of the following is correct in respect of the above function?
A.	f(x) is continuous at x = 0 but not differentiable at x = 0
B.	f(x) is continuous as well as differentiable at x = 0
C.	f(x) is discontinuous at x = 0
D.	None of the above
Answer» D. None of the above

Discussion

35.	Evaluate \(\lim_{x\rightarrow0} \frac {x\tan x}{1 - \cos x}\)
A.	1 / 2
B.	-1 / 2
C.	-2
D.	2
Answer» E.

Discussion

36.	At x = 0, the function \(f(x) =\left \| \frac{\sin2\pi x}{L}\right \|\) (- ∞ < x < ∞, L > 0) is
A.	Continuous and differentiable.
B.	Not continuous and not differentiable
C.	Not continuous but differentiable.
D.	Continuous but not differentiable.
Answer» E.

Discussion

37.	If a function f(x) = \(\begin{Bmatrix} \rm 3x + 2, x \geq1\\ \rm 5, \;\;\;\;\;\;\;\;x<1 \end{Bmatrix}\), then the function at x = 1
A.	Continuous and f(1) = 5
B.	Continuous but f(1) cannot be determined
C.	Not continuous
D.	None of the above
Answer» B. Continuous but f(1) cannot be determined

Discussion

38.	\(\mathop {\lim }\limits_{n \to \infty } \frac{{{1^{99}} + {2^{99}} + {3^{99}} + \ldots + {n^{99}}}}{{{n^{100}}}}\)
A.	99/100
B.	1/100
C.	1/99
D.	1/101
Answer» C. 1/99

Discussion

39.	Consider the following statements:1. If \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{f}}\left( {\rm{x}} \right)\) and \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{g}}\left( {\rm{x}} \right)\) both exist, then \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} \left\{ {{\rm{f}}\left( {\rm{x}} \right){\rm{g}}\left( {\rm{x}} \right)} \right\}\) exists.2. If \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} \left\{ {{\rm{f}}\left( {\rm{x}} \right){\rm{g}}\left( {\rm{x}} \right)} \right\}\) exists, then both \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{f}}\left( {\rm{x}} \right)\) and \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{g}}\left( {\rm{x}} \right)\) must exist.Which of the above statements is/are correct?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» B. 2 only

Discussion

40.	Find out the value of x if logx 4 + logx 16 + logx 64 = 12
A.	1
B.	2
C.	7
D.	54
Answer» C. 7

Discussion

41.	Let f(x) be a polynomial of degree four, having extreme value at x = 1 and x = 2.If \(\lim_{x \rightarrow 0} \left[1 + \frac {f(x)} {x^2} \right] = 3\) then f(2) is?
A.	0
B.	4
C.	-8
D.	-4
Answer» B. 4

Discussion

Explore topic-wise MCQs in Mathematics.

Consider the function f defined by \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {{x^2} - 1,}&{x < 3}\\ {2ax,}&{x \ge 3} \end{array}} \right.\) for all real numbers x. If f is continuous at x = 3, then value of a

Let f: R → R be defined by \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {x\sin \left( {\frac{1}{x}} \right)}&{if\;x > 0}\\ 0&{x \le 0} \end{array}} \right.\) Then

If \({\rm{F}}\left( {\rm{x}} \right) = \sqrt {9 - {{\rm{x}}^2}} \), then what is \(\mathop {\lim }\limits_{{\rm{x}} \to 1} \frac{{{\rm{F}}\left( {\rm{x}} \right) - {\rm{F}}\left( 1 \right)}}{{{\rm{x}} - 1}}\) equal to?

\(\frac{1}{{{{\log }_2}x}} + \frac{1}{{{{\log }_3}x}} + \frac{1}{{{{\log }_4}x}} + \ldots .. + \frac{1}{{{{\log }_{50}}x}},x \ne 1\) is equal to

If \(\rm \displaystyle\lim_{x \rightarrow 1} \dfrac{x^4-1}{x-1} = \lim_{x\rightarrow k} \dfrac{x^3-k^3}{x^2-k^2}\), where k ≠ 0, then what is the value of k?

If \({\rm{G}}\left( {\rm{x}} \right) = \sqrt {(25 - {{\rm{x}}^2}} \) then what is \(\mathop {\lim }\limits_{{\rm{x}} \to 1} \frac{{{\rm{G}}\left( {\rm{x}} \right) - {\rm{G}}\left( 1 \right)}}{{{\rm{x}} - 1}}{\rm{\;}}\)equal to?

\(\displaystyle\lim_{x\rightarrow 0} \dfrac{a^x-b^x}{e^x-1}\) is equal to :

If \({\rm{f}}\left( {\rm{x}} \right) = \frac{{{\rm{sin}}\left( {{{\rm{e}}^{{\rm{x}} - 2}} - 1} \right)}}{{{\rm{In}}\left( {{\rm{x}} - 1} \right)}}\), then \(\mathop {\lim }\limits_{{\rm{x}} \to 2} {\rm{f}}\left( {\rm{x}} \right)\) is equal to

\(\mathop {{\rm{lim}}}\limits_{x \to \infty } \left( {\frac{{2 + {x^2}}}{{1 + x\;}} - Ax - B} \right) = 3\)What is the value of B?

\(\mathop {{\rm{lim}}}\limits_{x \to \infty } \left( {\frac{{2 + {x^2}}}{{1 + x\;}} - Ax - B} \right) = 3\)What is the value of A?

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?

If \(\mathop {\lim }\limits_{{\rm{x}} \to \frac{{\rm{\pi }}}{2}} \frac{{\sin {\rm{x}}}}{{\rm{x}}} = {\rm{l}}\) and \(\mathop {\lim }\limits_{{\rm{x}} \to \infty } \frac{{\cos {\rm{x}}}}{{\rm{x}}} = {\rm{m}}\), then which one of the following is correct?

If \({\rm{f}}\left( {\rm{x}} \right) = \sqrt {25 - {{\rm{x}}^2}} {\rm{\;}},{\rm{\;}}\) then what is \(\mathop {{\rm{Lim}}}\limits_{{\rm{x}} \to 1} \frac{{{\rm{f}}\left( {\rm{x}} \right) - {\rm{f}}\left( 1 \right)}}{{{\rm{x}} - 1}}\) equal to?

If \(\mathop {\lim }\limits_{{\rm{x}} \to 0} \phi \left( {\rm{x}} \right) = {{\rm{a}}^2}\), where a ≠ 0, then what is \(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 0} \phi \left( {\frac{{\rm{x}}}{{\rm{a}}}} \right)\) Equal to

If f(x) =\(\left\lbrace \begin{matrix}\dfrac{\sin [\rm x]}{[\rm x]}, \ \ [\rm x] \neq 0 \\\ 0, \ \ [\rm x] = 0\end{matrix} \right.\), where [x] is the largest integer but not larger than x, then \(\rm \displaystyle\lim_{x \rightarrow 0} f(x)\) is

If \(x + \frac{1}{x} = \sqrt{3}\), then the value of x18 + x12 + x6 + 1 is

f(x) = x + |x| is continuous for

Consider the following statements for f(x) = e-|x| ;1. The function is continuous at x = 0.2. The function is differentiable at x = 0.Which of the above statements is / are correct?

Let f : A → R, where A = R\(0) is such that \({\rm{f}}\left( {\rm{x}} \right) = \frac{{{\rm{x}} + \left| {\rm{x}} \right|}}{{\rm{x}}}\). On which one of the following sets is f(x) continuous?

If \(\displaystyle\lim_{x\rightarrow \infty}\left(1+ \dfrac{a}{x}+\dfrac{b}{x^2}\right)^{2x}=e^2\), then the value of a and b are

If a + b + c = 5 and ab + bc + ca = 10, then the value of a3 + b3 + c3 - 3abc is

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)?

If \(\rm f(x) = \left \{ \begin{matrix} \rm x^2; & \rm x \leq 0 \\ \rm 2\sin x; & \rm x > 0 \end{matrix}\right.\), then x = 0 is a point of:

If \(\rm \lim_{x \to a} \frac{a^x -x^a}{x^x -a^a}= - 1\), then what is the value of a?

If \(\rm f(x) = \left\{ \begin{matrix} \rm \dfrac{x-x^2}{2x}; & \rm x \ne 0 \\ \rm K; & \rm x = 0 \end{matrix}\right.\) is a continuous function at x = 0, then the value of k is:

\(\lim_{x \rightarrow 3} \frac {\sqrt {3x} - 3}{\sqrt {2x - 4} - \sqrt 2}\) is equal to

Consider the following statements:1. f(x) = [x], where [.] is the greatest integer function, is discontinuous at x = n, where n ϵ Z.2. f(x) = cot x is discontinuous at x = nπ, where n ϵ Z. Which of the above statements is/are correct?

Find the value of alogaN

\(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {\frac{1}{n}} {e^{r/n}}\) is

If \(x + \frac{1}{x} = 2\), then the value of \({x^3} + \frac{1}{{{x^3}}}\) is

Let \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\cos \left[ x \right],}&{x \ge 0}\\ {\left| x \right| + a,}&{x < 0} \end{array}} \right.,\) where [x] denotes the greatest integer ≤ x. If f should be continuous at x = 0, then a must be

If \(\rm f(x)=\dfrac{\sin x}{x}\), where x ∈ R, is to be continuous at x = 0, then the value of the function at x = 0

A function is defined as follows:\({\rm{f}}\left( {\rm{x}} \right):\left\{ {\begin{array}{*{20}{c}} { - \frac{{\rm{x}}}{{\sqrt {{{\rm{x}}^2}} }},{\rm{\;x}} \ne 0}\\ {0.{\rm{\;x}} = 0} \end{array}} \right.\)Which one of the following is correct in respect of the above function?

Evaluate \(\lim_{x\rightarrow0} \frac {x\tan x}{1 - \cos x}\)

At x = 0, the function \(f(x) =\left | \frac{\sin2\pi x}{L}\right |\) (- ∞ < x < ∞, L > 0) is

If a function f(x) = \(\begin{Bmatrix} \rm 3x + 2, x \geq1\\ \rm 5, \;\;\;\;\;\;\;\;x<1 \end{Bmatrix}\), then the function at x = 1

\(\mathop {\lim }\limits_{n \to \infty } \frac{{{1^{99}} + {2^{99}} + {3^{99}} + \ldots + {n^{99}}}}{{{n^{100}}}}\)

Find out the value of x if logx 4 + logx 16 + logx 64 = 12

Let f(x) be a polynomial of degree four, having extreme value at x = 1 and x = 2.If \(\lim_{x \rightarrow 0} \left[1 + \frac {f(x)} {x^2} \right] = 3\) then f(2) is?