183 + Interview Questions in CONTINUITY IN GENERAL AWARENESS Page 2 InterviewSolution

51.	The function `f(x)=(4-x^2)/(4x-x^3)`a)discontinuous at only one pointb) discontinuous exactly at two pointsc)discontinuous exactly at three pointsd) none of these
Answer» `f(x) = (4-x^2)/(4x-x^3 )` `= (4 -x^2)/(x(4- x^2))` `f(x) = (4-x^2)/(x(2-x)(2+x))` `x= 0, 2,-2` at `x=0,2,-2` so, f(x) is discontinuous answer

Discussion

52.	The function `f(x) =(2x^(2)+7)/(x^(3)+3x^(2)-x-3)` is discontinuous forA. `x=1` onlyB. `x=1, -1` onlyC. `x=1, -1,-3` and other values of xD. `x=1, -1, -3` only
Answer» Correct Answer - D

Discussion

53.	Determine the value of `k`for which the following function is continuous at `x=3.``f(x)={(x^2-9)/(x-3)` ,`x!=3`and k when `x=3`
Answer» `1) LHL = RHL` `2) LHL = RHL`= value of f(x) LHL=`lim_(x->3) (x^2 - 9)/(x-3)` `x= 3-h` `:. lim_(h->0) ((3-h)^2 - 9)/((3-h)-3)` `= lim_(h->0) (9-h^2 - 6h - 9)/((3-h) - 3)` `= lim_(h->0) (h(h-6))/(-1)` `= 6` RHL=`lim_(x->3) (x^2-9)/(x-3)` `x= 3 + h` `lim_(h->0) ((3+h)^2 - 9)/(3+h-3)` `lim_(h->0) (h^2 + 6h)/h` `lim_(h->0) h + 6 = 6` LHL=RHL = f(3) `6= k` Answer

Discussion

54.	Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = e
Answer» Given, f (x) = e It is a constant function …. [∵ e = 2.71828] ∴ f(x) is continuous on the set of real numbers i.e., x ∈ R.

54.

Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = e

Answer»

Given, f (x) = e

It is a constant function …. [∵ e = 2.71828]

∴ f(x) is continuous on the set of real numbers i.e., x ∈ R.

Discussion

55.	Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = 7
Answer» Given, f(x) = 7 It is a constant function. ∴ f(x) is continuous on the set of real numbers i.e., x ∈ R.

55.

Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = 7

Answer»

Given, f(x) = 7

It is a constant function.

∴ f(x) is continuous on the set of real numbers i.e., x ∈ R.

Discussion

56.	Are the following functions continuous on the set of real numbers? Justify your answers.f (x) = log 19
Answer» Given, f(x) = log 19 Here, log 19 is a constant ∴ f(x) is a constant function ∴ f(x) is continuous on the set of real numbers i.e., x ∈ R.

56.

Are the following functions continuous on the set of real numbers? Justify your answers.f (x) = log 19

Answer»

Given, f(x) = log 19

Here, log 19 is a constant

∴ f(x) is a constant function

∴ f(x) is continuous on the set of real numbers i.e., x ∈ R.

Discussion

57.	g(x) = sin (4x - 3)
Answer» Given, g(x) = sin (4x - 3) It is a sine function ∴ f(x) is continuous on the set of real numbers i.e., x ∈ R.

57.

g(x) = sin (4x - 3)

Answer»

Given, g(x) = sin (4x - 3)

It is a sine function

∴ f(x) is continuous on the set of real numbers i.e., x ∈ R.

Discussion

58.	Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = 5x
Answer» Given, f(x) = 5^x It is an exponential function ∴ It is continuous on the set of real numbers i.e., x ∈ R.

58.

Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = 5x

Answer»

Given, f(x) = 5^x

It is an exponential function

∴ It is continuous on the set of real numbers i.e., x ∈ R.

Discussion

59.	Determine the value of constant k so that the function\(f(x) =\begin{cases}\frac{1 - cos\,kx}{x\,sin\,x}, & \quad x \neq 0\\2, & \quad x=0\end{cases}\) is continuous at x = 0f(x) = {(1 - cos kx)/(x sin x), x ≠ 0, 2, x = 0 is continuous at x = 0(A) k = ± 2(B) k = ± 4(C) k = ± 1(D) k = ± 3
Answer» Correct answer is (A) k = ± 2

59.

Determine the value of constant k so that the function\(f(x) =\begin{cases}\frac{1 - cos\,kx}{x\,sin\,x}, & \quad x \neq 0\\2, & \quad x=0\end{cases}\) is continuous at x = 0f(x) = {(1 - cos kx)/(x sin x), x ≠ 0, 2, x = 0 is continuous at x = 0(A) k = ± 2(B) k = ± 4(C) k = ± 1(D) k = ± 3

Answer»

Correct answer is

(A) k = ± 2

Discussion

60.	Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = e(5x + 7)
Answer» Given, f(x) = e^{(5x + 7)} It is an exponential function ∴ It is continuous on the set of real numbers i.e., x ∈ R.

60.

Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = e(5x + 7)

Answer»

Given, f(x) = e^{(5x + 7)}

It is an exponential function

∴ It is continuous on the set of real numbers i.e., x ∈ R.

Discussion

61.	The function \(f(x) = \frac{log(1 + ax) - log(1 - bx)}{x}\) is not defined at x = 0. The value which should be assigned to f at x = 0 so that it is continuous at x = 0, isf(x) = (log(1 + ax) - log(1 - bx))/(x)(A) a - b(B) a + b(C) log a + log b(D) log a - log b
Answer» Correct answer is (B) a + b

61.

The function \(f(x) = \frac{log(1 + ax) - log(1 - bx)}{x}\) is not defined at x = 0. The value which should be assigned to f at x = 0 so that it is continuous at x = 0, isf(x) = (log(1 + ax) - log(1 - bx))/(x)(A) a - b(B) a + b(C) log a + log b(D) log a - log b

Answer»

Correct answer is

(B) a + b

Discussion

62.	Let`f(x)={(1-cos4x)/(x^2a,),ifx0,ifx=0`Determine the value of `a`so that `f(x)`is continuous at `x=0.`
Answer» LHL=`f(0^-)` `f(x) = (1- cos 4x)/x^2` `= (2sin^2 2x)/(x^2)` `=> 2(sin2x)/(2x) (sin2x)/(2x) 2*2= 8` `sqrtx/(sqrt(16 +sqrtx) - 4) xx (sqrt(16+ sqrtx))/(sqrt(16 + sqrtx) + 4)` `=(sqrtx( sqrt(16 + sqrtx) + 4))/(16 + sqrtx - 16)` `= sqrt(16 + sqrtx) + 4` `= sqrt16 + 4 = 4 + 4 = 8` so,`a=8` Answer

Discussion

63.	Given the function `f(x)=1/(x+2)`. Find the points of discontinuity of the function `f(f(x))`
Answer» `f(x) = 1/(x+2) ` `x=-2` `f(f(x)) = 1/((1/(x+2)) + 2)` `= (x+2)/(2x+5)` `2x+5= 0` `x=-5/2` Answer

Discussion

64.	For what value of `k`is the function`f(x)={(x^2-1)/(x-1)k,``,x!=1,x=1"c o n t i n u o u sa t"x=1?`
Answer» As `f(x)` is continuous at `x = 1`, `:. Lim_(x->0) f(x) = f(1).` `=>Lim_(x->0)(x^2-1)/(x-1) = k` `=>Lim_(x->0)((x+1)(x-1))/(x-1) = k` `=>Lim_(x->0)(x+1) = k` `=> 2 = k` `:.` For `k = 2`, given function will be continuous at `x = 1`.

Discussion

65.	Let`f(x)=(log(1+x/a)-log(1-x/b))/x ,x!=0.`Find the value of `f` at `x=0`so that `f`becomes continuous at `x=0`
Answer» `f(x) = (log(1) - log(1))/0 = 0/0` `f(0) = f(0+h) = f(0-h) ` `l = lim_(x->0) f(x) = lim(x->0) (log(1 + x/a) - log(1-x/b))/x = 0/0` applying l hospital rule `l = lim_(x->0) (1/(1+ x/a) 1/a - 1/(1-x/b) (-1/b))/1` `l = lim_(x->0) ((1/(1+0) 1/a + (1/(1-0) *1/b))/1)` `l = 1/a + 1/b = (a+b)/(ab)` so, `f(0) = (a+b)/(ab)` Answer

Discussion

66.	If the function `f(x)`defined byf(x)=`(log(1+3x)-"log"(1-2x))/x `, `x!=0` and k , x=0.Find k.
Answer» `f(x) = (log(1+3x) - log(1-2x))/x` `l = lim_(x->0) (log(1+3x) - log(1-2x))/x` `l = lim_(x->0) (1/(1+3x) * 3 - 1/(1-2x) (-2))/1` `l = lim_(x->0) 3/(1+3x) + 2/(1-2x) ` `l = 3/1 + 2/1 = 5` `f(0) = f(0^-) = f(0^+)` `k = 5` Answer

Discussion

67.	The value of `f(0),`so that the function`f(x)=(sqrt(a^2-a x+x^2)-sqrt(a^2+a x+x^2))/(sqrt(a+x)-sqrt(a-x))`becomes continuous for all `x ,`given by`a^(3/2)`(b) `a^(1/2)`(c) `-a^(1/2)`(d) `-a^(3/2)`A. `-a sqrt(a)`B. `a sqrt(a)`C. `-sqrt(a)`D. `sqrt(a)`
Answer» Correct Answer - C

Discussion

68.	The function\(f(x) =\begin{cases}3x - 5, & \quad \text{for}\, x<3\\x+1, & \quad \text{for}\, x>3\\c, & \quad \text{for}\, x=3\end{cases}\) is continuous at x = 3, if c is equal tof(x) = {(3x -5), for x < 3, (x + 1), for x > 3, c, for x = 3(A) 4(B) 3(C) 1(D) 2
Answer» Correct answer is (A) 4

68.

The function\(f(x) =\begin{cases}3x - 5, & \quad \text{for}\, x<3\\x+1, & \quad \text{for}\, x>3\\c, & \quad \text{for}\, x=3\end{cases}\) is continuous at x = 3, if c is equal tof(x) = {(3x -5), for x < 3, (x + 1), for x > 3, c, for x = 3(A) 4(B) 3(C) 1(D) 2

Answer»

Correct answer is

(A) 4

Discussion

69.	Determine `f(0)`so that the function `f(x)`defined by`f(x)=((4^x-1)^3)/(sinx/4log(1+(x^2)/3))`becomes continuous at `x=0`
Answer» `lim_(x->0) ((4^x - 1)^3)/(sin(x/4) log(1 + x^2/3))` `(4^x - 1)^3= (e^(x ln4) - 1)^3 = (1 + xln 4 + (xln4)^2/2 ....-1)^3` `e^t = 1 + t + t^2/2 + .......` or `sin(x/4)/(x/4) [ log(1 + x^2/3)] xx x/4 ` or `= [x/3 + (x/3)^2 /2+ ....] xx pi/4` now, `([xln 4 + (xln4)^2/2]^3)/(x/4* [(x^2/3)^2/2])` `= (x^3(ln^3 4)/((x/4)(x^2/3)) = 12 (ln4)^3` `f(0) = 12( ln 4)^3` Answer

Discussion

70.	Show that`f(x)=5x-4` , `0< x < 1``f(x)=4x^3-3x` , `1< x
Answer» `x<= 1` `f(x) = 5x- 4` `f(1^-) = 5 xx 1 - 4 =1` `x > 1` `f(x) = 4x^23 - 3x` `f(1^+) = 4 xx1 - 3(1) = 1` `f(1^-) = f(1^+)` `:. ` continuous

Discussion

71.	If`f(x)={asinpi/2(x+1), ,xlt=0(tanx-sinx)/(x^3),x >0`is continuous at `x=0`, then a equal(a)`1/2`(b) `1/3`(c) `1/4`(d) `1/6`
Answer» `f(x) = { a sin pi/2 ( x+1) x<= 0 ` `= { (tanx - sin x)/x^3 x>0` `f(0^-) = (tanx- sin x)/x^3` `= lim_(x->0) (tan x - sinx)/x^3` `= lim_(x->0) (sinx[1/cosx - 1])/x^3` `= lim_(x-0) (1-cos x)/(x^2 cos x)` `= lim_(x->0) (1-(1- 2sin^2 (x/2)))/(x^2 cos x)` `= lim_(x->0) (2sin^2 (x/2))/(4 (x/2)^2 cosx)` `= lim_(x->0) 1/(2cos x) = 1/2` `a=1/2` option a is correct

Discussion

72.	If `f(x)` is continuous at `x=pi/4`, where `f(x)=(1-tanx)/(1-sqrt(2)sin x)`, for `x!= pi/4`, then `f(pi/4)=`A. 2B. `sqrt(2)`C. `2sqrt(2)`D. `(1)/(sqrt(2))`
Answer» Correct Answer - A

Discussion

73.	If \(f(x) =\begin{cases}\frac{sin\,3x}{sin\,x}, & x \neq 0\\k, & x=0\end{cases}\) is a continuous function, then k =f(x) = {(sin 3x)/(sin x), x ≠ 0, k, x = 0(A) 1(B) 3(C) \(\frac{1}{3}\)(D) 0
Answer» Correct answer is (B) 3

73.

If \(f(x) =\begin{cases}\frac{sin\,3x}{sin\,x}, & x \neq 0\\k, & x=0\end{cases}\) is a continuous function, then k =f(x) = {(sin 3x)/(sin x), x ≠ 0, k, x = 0(A) 1(B) 3(C) \(\frac{1}{3}\)(D) 0

Answer»

Correct answer is

(B) 3

Discussion

74.	At what point on the curve y = x2 , x ∈ [-2, 2] at which the tangent is parallel to x-axis?
Answer» Y = x² , a continuous function on [-2, 2] and dierentiable on [-2, 2] f(2) = 4 = f(-2). All conditions of Rolles theorem is satised. Given the tangent is parallel to x-axis. f ^-1 (x) = 2x f^-1 (c) = 2c f^-1 (c) = 0 ⇒ 2c = 0 ⇒ c = 0 ∈ [-2, 2] where c = 0, y = 0 Therefore (0, 0) is the required point.

74.

At what point on the curve y = x2 , x ∈ [-2, 2] at which the tangent is parallel to x-axis?

Answer»

Y = x² , a continuous function on [-2, 2] and dierentiable on [-2, 2] f(2) = 4 = f(-2).

All conditions of Rolles theorem is satised.

Given

the tangent is parallel to x-axis.

f ^-1 (x) = 2x

f^-1 (c) = 2c

f^-1 (c) = 0 ⇒ 2c = 0 ⇒ c = 0 ∈ [-2, 2]

where c = 0, y = 0

Therefore (0, 0) is the required point.

Discussion

75.	If \(f(x) = \begin{cases} \frac{sin(a + 1)x + sin\,x}{x}, & \quad x<0\\ \frac{1}{2}, & \quad x=0\\\frac{x^{3\sqrt{2}+1}}{2}, & \quad x>0 \end{cases}\) is continuous at x = 0, then the value of a isIf f(x) = {(sin(a + 1)x + sin x)/x, x<0, 1/2, x = 0, (x^3√2 + 1)/2, x > 0, then the value of a isA. \(\frac{1}{2}\)B. \(-\frac{1}{2}\)C. \(\frac{3}{2}\)D. \(-\frac{3}{2}\)
Answer» Correct answer is (D) \(-\frac{3}{2}\)

75.

If \(f(x) = \begin{cases} \frac{sin(a + 1)x + sin\,x}{x}, & \quad x<0\\ \frac{1}{2}, & \quad x=0\\\frac{x^{3\sqrt{2}+1}}{2}, & \quad x>0 \end{cases}\) is continuous at x = 0, then the value of a isIf f(x) = {(sin(a + 1)x + sin x)/x, x<0, 1/2, x = 0, (x^3√2 + 1)/2, x > 0, then the value of a isA. \(\frac{1}{2}\)B. \(-\frac{1}{2}\)C. \(\frac{3}{2}\)D. \(-\frac{3}{2}\)

Answer»

Correct answer is

(D) \(-\frac{3}{2}\)

Discussion

76.	Function \(f(x) = \begin{cases} \frac{x^2 - 4x + 3}{x^2 - 1}, & \quad x\neq 1\\ 2, & \quad x=1 \end{cases},\) isf(x) = {(x2 - 4x + 3)/(x2 - 1), x ≠ 1, 2, x = 1 is(A) continuous at x = 1.(B) continuous at x = -1.(C) continuous at x = 1 and x = -1.(D) discontinuous at x = 1.
Answer» Correct answer is (D). discontinuous at x = 1.

76.

Function \(f(x) = \begin{cases} \frac{x^2 - 4x + 3}{x^2 - 1}, & \quad x\neq 1\\ 2, & \quad x=1 \end{cases},\) isf(x) = {(x2 - 4x + 3)/(x2 - 1), x ≠ 1, 2, x = 1 is(A) continuous at x = 1.(B) continuous at x = -1.(C) continuous at x = 1 and x = -1.(D) discontinuous at x = 1.

Answer»

Correct answer is

(D). discontinuous at x = 1.

Discussion

77.	If the function `f(x) {((1 - cos 4x)/(8x^(2))",",x !=0),(" k,",x = 0):}`is continuous at x = 0 then k = ?A. 16B. 2C. -1D. 1
Answer» Correct Answer - D

Discussion

78.	If `f(x) ` is continuous at `x=0`, where `f(x)={:{((9^(x)-9^(-x))/(sin x)", for " x!=0),(k", for " x=0):}`, then `k=`A. log 9B. log 81C. 2 log 3D. `(log 9)^(2)`
Answer» Correct Answer - B

Discussion

79.	If `f(x)` is continuous at `x=0`, where `f(x)=(10^(x)+7^(x)-14^(x)-5^(x))/(1-cos 4x)`, for `x!=0`, then `f(0)=`A. `1/4 (log2)log(5/7)`B. `1/8 (log2) log(5/7)`C. `1/4 (log2) log(7/5)`D. `1/8 (log 2) log (7/5)`
Answer» Correct Answer - B

Discussion

80.	The function f(x) is condtions at the point x=0 where `f(x)=log(1+kx)/(sinx), "for" x ne 0 ` = 5 for x=0 then value of k isA. 5B. -5C. `1/5`D. `(-1)/(5)`
Answer» Correct Answer - A

Discussion

81.	If `f(x)` is continuous at `x=0`, where `f(x)=((5^(x)-2^(x))x)/(cos 5x-cos 3x)`, for `x!=0`, then `f(0)=`A. `(-1)/(4) log (2/5)`B. `1/4 log (2/5)`C. `(-1)/(8) log (2/5)`D. `1/8 log (2/5)`
Answer» Correct Answer - D

Discussion

82.	If `f(x)` is continuous at `x=pi/2`, where `f(x)=(sqrt(2)-sqrt(1+sin x))/(cos^(2)x)`, for `x!= pi/2`, then `f(pi/2)=`A. `4sqrt(2)`B. `2sqrt(2)`C. `(1)/(4sqrt(2))`D. `(1)/(2sqrt(2))`
Answer» Correct Answer - C

Discussion

83.	If `f(x)` is continuous at `x=0`, where `f(x)=(4^(x)-2^(x+1)+1)/(1-cos x)`, for `x!=0`, then `f(0)=`A. `(2 log2)^(2)`B. `2(log 2)^(2)`C. `(log2)^(2)`D. `((log2)^(2))/(2)`
Answer» Correct Answer - B

Discussion

84.	If `f(x)` is continuous at `x=a,a>0`, where `f(x)=[(a^x-x^a)/(x^x-a^a)`, for `x!=a`, -1 for x=a, then `a=`A. eB. 2eC. 1D. 0
Answer» Correct Answer - C

Discussion

85.	Evaluate :`("lim")_(xvec2a^+)(sqrt(x-2a)+sqrt(x)-sqrt(2a))/(x^2-4a^2)`A. `2sqrt(a)`B. 2aC. `(1)/(2sqrt(a))`D. `(1)/(2a)`
Answer» Correct Answer - C

Discussion

86.	If the function f(x) defined by `f(x)={(x"sin"(1)/(x)",", "for "x ne 0),(k",", "for " x=0):}` is continuous at x = 0, then k is equal to
Answer» Correct Answer - A

Discussion

87.	If `f(x)` is continuous at `x=a`, where `f(x)=(sin (a+x)+sin(a-x)-2 sin a)/(x sin x)", for " x!=a`, then `f(a)=`A. `2/a (cosa-1)`B. `1/a (cos a-1)`C. `1/a (1-cosa)`D. None of these
Answer» Correct Answer - A

Discussion

88.	If `f(x)` is continuous at `x=sqrt(2)`, where `f(x)=(sqrt(3+2x)-(sqrt(2)+1))/(x^(2)-2)`, for `x!= sqrt(2)`, then `f(sqrt(2))=`A. `(1)/(2(2+sqrt(2)))`B. `(1)/(sqrt(2)(2+sqrt(2)))`C. `(1)/(2+sqrt(2))`D. `(1)/(2+2sqrt(2))`
Answer» Correct Answer - A

Discussion

89.	If `f(x)` is continuous at `x=0`, where `f(x)=sin x-cos x`, for `x!=0`, then `f(0)=`A. 2B. 0C. -1D. 1
Answer» Correct Answer - C

Discussion

90.	If `f(x)` is continuous at `x=0`, where `f(x)=(3-4cos x+cos 2x)/(x^(2))`, for `x!=0`, then `f(0)=`
Answer» Correct Answer - A

Discussion

91.	The value of k which makes `f(x)={:{(sin(1/x)", for " x!=0),(k", for " x=0):}` continuous at `x=0` is
Answer» Correct Answer - D

Discussion

92.	If `f(x)` is continuous at `x=0`, where `f(x)=(3-4cos x+cos 2x)/(x^(4))`, for `x!=0`, then `f(0)=`A. `1/4`B. `1/2`C. 8D. 4
Answer» Correct Answer - B

Discussion

93.	If `f(x)` is continuous at `x=5`, where `f(x)=(sqrt(3+sqrt(4+x))-sqrt(6))/(x-5)`, for `x!=5`, then `f(5)=`A. `(1)/(2sqrt(6))`B. `(1)/(3sqrt(6))`C. `(1)/(12sqrt(6))`D. None of these
Answer» Correct Answer - C

Discussion

94.	If `f(x)` is continuous at `x=-2`, where `f(x)=(2)/(x+2)+(1)/(x^(2)-2x+4)-(24)/(x^(3)+8)`, for `x!= -2`, then `f(-2)=`A. `(-1)/(4)`B. `1/4`C. `(11)/(12)`D. `(-11)/(12)`
Answer» Correct Answer - D

Discussion

95.	If `f(x)` is continuous at `x=2`, where `f(x)=((x^(2)-x-2)^(20))/((x^(3)-12x+16)^(10))`, for `x!=2`, then `f(2)=`A. `(3^(20))/(2^(10))`B. `(3^(10))/(2^(20))`C. `(3/2)^(10)`D. `(3/2)^(20)`
Answer» Correct Answer - C

Discussion

96.	If `f(x)` is contiuous at `x= pi/2`, where `f(x)=(cosec x - sin x)/(pi/2 - x)", for " x != pi/2`, then `f(pi/2)=`A. `1/4`B. `1/2`C. `1/6`D. `1/8`
Answer» Correct Answer - B

Discussion

97.	if the function `f(x)=(x^2-(a+2)x+a)/(x-2)` for `x!=2` and `f(x)=2` for `x=2` is continuous function at `x=2` then value of` a` is:A. 2B. -1C. 1D. 0
Answer» Correct Answer - D

Discussion

98.	If `f(x)` is continuous at `x=pi/2`, where `f(x)=(3^(x- pi/2)-6^(x- pi/2))/(cos x)` , for `x != pi/2`, then `f(pi/2)=`A. log 3B. log 6C. log 2D. log 18
Answer» Correct Answer - C

Discussion

99.	If `f(x)` is continuous at `x = pi`, where `f(x)=(sqrt(2+cos x)-1)/((pi-x)^(2))", for " x!= pi`, then `f(pi)=`A. `1/4`B. `(-1)/(4)`C. `1/2`D. `(-1)/(2)`
Answer» Correct Answer - A

Discussion

100.	If `f(x)` is continuous for all x, where `f(x)={:{((x^(2)-7x+12)/((x-2)^(2))", for "x!=2),(k ", for" x = 2 ):}`, then `k=`A. 7B. -7C. `pm7`D. None of these
Answer» Correct Answer - A

Discussion

Explore topic-wise InterviewSolutions in .

The function `f(x)=(4-x^2)/(4x-x^3)`a)discontinuous at only one pointb) discontinuous exactly at two pointsc)discontinuous exactly at three pointsd) none of these

The function `f(x) =(2x^(2)+7)/(x^(3)+3x^(2)-x-3)` is discontinuous forA. `x=1` onlyB. `x=1, -1` onlyC. `x=1, -1,-3` and other values of xD. `x=1, -1, -3` only

Determine the value of `k`for which the following function is continuous at `x=3.``f(x)={(x^2-9)/(x-3)` ,`x!=3`and k when `x=3`

Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = e

Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = 7

Are the following functions continuous on the set of real numbers? Justify your answers.f (x) = log 19

g(x) = sin (4x - 3)

Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = 5x

Are the following functions continuous on the set of real numbers? Justify your answers.f(x) = e(5x + 7)

The function \(f(x) = \frac{log(1 + ax) - log(1 - bx)}{x}\) is not defined at x = 0. The value which should be assigned to f at x = 0 so that it is continuous at x = 0, isf(x) = (log(1 + ax) - log(1 - bx))/(x)(A) a - b(B) a + b(C) log a + log b(D) log a - log b

Let`f(x)={(1-cos4x)/(x^2a,),ifx0,ifx=0`Determine the value of `a`so that `f(x)`is continuous at `x=0.`

Given the function `f(x)=1/(x+2)`. Find the points of discontinuity of the function `f(f(x))`

For what value of `k`is the function`f(x)={(x^2-1)/(x-1)k,``,x!=1,x=1"c o n t i n u o u sa t"x=1?`

Let`f(x)=(log(1+x/a)-log(1-x/b))/x ,x!=0.`Find the value of `f` at `x=0`so that `f`becomes continuous at `x=0`

If the function `f(x)`defined byf(x)=`(log(1+3x)-"log"(1-2x))/x `, `x!=0` and k , x=0.Find k.

The value of `f(0),`so that the function`f(x)=(sqrt(a^2-a x+x^2)-sqrt(a^2+a x+x^2))/(sqrt(a+x)-sqrt(a-x))`becomes continuous for all `x ,`given by`a^(3/2)`(b) `a^(1/2)`(c) `-a^(1/2)`(d) `-a^(3/2)`A. `-a sqrt(a)`B. `a sqrt(a)`C. `-sqrt(a)`D. `sqrt(a)`

The function\(f(x) =\begin{cases}3x - 5, &amp; \quad \text{for}\, x&lt;3\\x+1, &amp; \quad \text{for}\, x&gt;3\\c, &amp; \quad \text{for}\, x=3\end{cases}\) is continuous at x = 3, if c is equal tof(x) = {(3x -5), for x &lt; 3, (x + 1), for x &gt; 3, c, for x = 3(A) 4(B) 3(C) 1(D) 2

Determine `f(0)`so that the function `f(x)`defined by`f(x)=((4^x-1)^3)/(sinx/4log(1+(x^2)/3))`becomes continuous at `x=0`

Show that`f(x)=5x-4` , `0< x < 1``f(x)=4x^3-3x` , `1< x

If`f(x)={asinpi/2(x+1), ,xlt=0(tanx-sinx)/(x^3),x >0`is continuous at `x=0`, then a equal(a)`1/2`(b) `1/3`(c) `1/4`(d) `1/6`

If `f(x)` is continuous at `x=pi/4`, where `f(x)=(1-tanx)/(1-sqrt(2)sin x)`, for `x!= pi/4`, then `f(pi/4)=`A. 2B. `sqrt(2)`C. `2sqrt(2)`D. `(1)/(sqrt(2))`

If \(f(x) =\begin{cases}\frac{sin\,3x}{sin\,x}, &amp; x \neq 0\\k, &amp; x=0\end{cases}\) is a continuous function, then k =f(x) = {(sin 3x)/(sin x), x ≠ 0, k, x = 0(A) 1(B) 3(C) \(\frac{1}{3}\)(D) 0

At what point on the curve y = x2 , x ∈ [-2, 2] at which the tangent is parallel to x-axis?

Function \(f(x) = \begin{cases} \frac{x^2 - 4x + 3}{x^2 - 1}, &amp; \quad x\neq 1\\ 2, &amp; \quad x=1 \end{cases},\) isf(x) = {(x2 - 4x + 3)/(x2 - 1), x ≠ 1, 2, x = 1 is(A) continuous at x = 1.(B) continuous at x = -1.(C) continuous at x = 1 and x = -1.(D) discontinuous at x = 1.

If the function `f(x) {((1 - cos 4x)/(8x^(2))",",x !=0),(" k,",x = 0):}`is continuous at x = 0 then k = ?A. 16B. 2C. -1D. 1

If `f(x) ` is continuous at `x=0`, where `f(x)={:{((9^(x)-9^(-x))/(sin x)", for " x!=0),(k", for " x=0):}`, then `k=`A. log 9B. log 81C. 2 log 3D. `(log 9)^(2)`

If `f(x)` is continuous at `x=0`, where `f(x)=(10^(x)+7^(x)-14^(x)-5^(x))/(1-cos 4x)`, for `x!=0`, then `f(0)=`A. `1/4 (log2)log(5/7)`B. `1/8 (log2) log(5/7)`C. `1/4 (log2) log(7/5)`D. `1/8 (log 2) log (7/5)`

The function f(x) is condtions at the point x=0 where `f(x)=log(1+kx)/(sinx), "for" x ne 0 ` = 5 for x=0 then value of k isA. 5B. -5C. `1/5`D. `(-1)/(5)`

If `f(x)` is continuous at `x=0`, where `f(x)=((5^(x)-2^(x))x)/(cos 5x-cos 3x)`, for `x!=0`, then `f(0)=`A. `(-1)/(4) log (2/5)`B. `1/4 log (2/5)`C. `(-1)/(8) log (2/5)`D. `1/8 log (2/5)`

If `f(x)` is continuous at `x=pi/2`, where `f(x)=(sqrt(2)-sqrt(1+sin x))/(cos^(2)x)`, for `x!= pi/2`, then `f(pi/2)=`A. `4sqrt(2)`B. `2sqrt(2)`C. `(1)/(4sqrt(2))`D. `(1)/(2sqrt(2))`

If `f(x)` is continuous at `x=0`, where `f(x)=(4^(x)-2^(x+1)+1)/(1-cos x)`, for `x!=0`, then `f(0)=`A. `(2 log2)^(2)`B. `2(log 2)^(2)`C. `(log2)^(2)`D. `((log2)^(2))/(2)`

If `f(x)` is continuous at `x=a,a>0`, where `f(x)=[(a^x-x^a)/(x^x-a^a)`, for `x!=a`, -1 for x=a, then `a=`A. eB. 2eC. 1D. 0

Evaluate :`("lim")_(xvec2a^+)(sqrt(x-2a)+sqrt(x)-sqrt(2a))/(x^2-4a^2)`A. `2sqrt(a)`B. 2aC. `(1)/(2sqrt(a))`D. `(1)/(2a)`

If the function f(x) defined by `f(x)={(x"sin"(1)/(x)",", "for "x ne 0),(k",", "for " x=0):}` is continuous at x = 0, then k is equal to

If `f(x)` is continuous at `x=a`, where `f(x)=(sin (a+x)+sin(a-x)-2 sin a)/(x sin x)", for " x!=a`, then `f(a)=`A. `2/a (cosa-1)`B. `1/a (cos a-1)`C. `1/a (1-cosa)`D. None of these

If `f(x)` is continuous at `x=sqrt(2)`, where `f(x)=(sqrt(3+2x)-(sqrt(2)+1))/(x^(2)-2)`, for `x!= sqrt(2)`, then `f(sqrt(2))=`A. `(1)/(2(2+sqrt(2)))`B. `(1)/(sqrt(2)(2+sqrt(2)))`C. `(1)/(2+sqrt(2))`D. `(1)/(2+2sqrt(2))`

If `f(x)` is continuous at `x=0`, where `f(x)=sin x-cos x`, for `x!=0`, then `f(0)=`A. 2B. 0C. -1D. 1

If `f(x)` is continuous at `x=0`, where `f(x)=(3-4cos x+cos 2x)/(x^(2))`, for `x!=0`, then `f(0)=`

The value of k which makes `f(x)={:{(sin(1/x)", for " x!=0),(k", for " x=0):}` continuous at `x=0` is

If `f(x)` is continuous at `x=0`, where `f(x)=(3-4cos x+cos 2x)/(x^(4))`, for `x!=0`, then `f(0)=`A. `1/4`B. `1/2`C. 8D. 4

If `f(x)` is continuous at `x=5`, where `f(x)=(sqrt(3+sqrt(4+x))-sqrt(6))/(x-5)`, for `x!=5`, then `f(5)=`A. `(1)/(2sqrt(6))`B. `(1)/(3sqrt(6))`C. `(1)/(12sqrt(6))`D. None of these

If `f(x)` is continuous at `x=-2`, where `f(x)=(2)/(x+2)+(1)/(x^(2)-2x+4)-(24)/(x^(3)+8)`, for `x!= -2`, then `f(-2)=`A. `(-1)/(4)`B. `1/4`C. `(11)/(12)`D. `(-11)/(12)`

If `f(x)` is continuous at `x=2`, where `f(x)=((x^(2)-x-2)^(20))/((x^(3)-12x+16)^(10))`, for `x!=2`, then `f(2)=`A. `(3^(20))/(2^(10))`B. `(3^(10))/(2^(20))`C. `(3/2)^(10)`D. `(3/2)^(20)`

If `f(x)` is contiuous at `x= pi/2`, where `f(x)=(cosec x - sin x)/(pi/2 - x)", for " x != pi/2`, then `f(pi/2)=`A. `1/4`B. `1/2`C. `1/6`D. `1/8`

if the function `f(x)=(x^2-(a+2)x+a)/(x-2)` for `x!=2` and `f(x)=2` for `x=2` is continuous function at `x=2` then value of` a` is:A. 2B. -1C. 1D. 0

If `f(x)` is continuous at `x=pi/2`, where `f(x)=(3^(x- pi/2)-6^(x- pi/2))/(cos x)` , for `x != pi/2`, then `f(pi/2)=`A. log 3B. log 6C. log 2D. log 18

If `f(x)` is continuous at `x = pi`, where `f(x)=(sqrt(2+cos x)-1)/((pi-x)^(2))", for " x!= pi`, then `f(pi)=`A. `1/4`B. `(-1)/(4)`C. `1/2`D. `(-1)/(2)`

If `f(x)` is continuous for all x, where `f(x)={:{((x^(2)-7x+12)/((x-2)^(2))", for "x!=2),(k ", for" x = 2 ):}`, then `k=`A. 7B. -7C. `pm7`D. None of these

The function\(f(x) =\begin{cases}3x - 5, & \quad \text{for}\, x<3\\x+1, & \quad \text{for}\, x>3\\c, & \quad \text{for}\, x=3\end{cases}\) is continuous at x = 3, if c is equal tof(x) = {(3x -5), for x < 3, (x + 1), for x > 3, c, for x = 3(A) 4(B) 3(C) 1(D) 2

If \(f(x) =\begin{cases}\frac{sin\,3x}{sin\,x}, & x \neq 0\\k, & x=0\end{cases}\) is a continuous function, then k =f(x) = {(sin 3x)/(sin x), x ≠ 0, k, x = 0(A) 1(B) 3(C) \(\frac{1}{3}\)(D) 0

Function \(f(x) = \begin{cases} \frac{x^2 - 4x + 3}{x^2 - 1}, & \quad x\neq 1\\ 2, & \quad x=1 \end{cases},\) isf(x) = {(x2 - 4x + 3)/(x2 - 1), x ≠ 1, 2, x = 1 is(A) continuous at x = 1.(B) continuous at x = -1.(C) continuous at x = 1 and x = -1.(D) discontinuous at x = 1.