The value of f(0), such that f(x)=(1/x2)(1- cos ( sin x)) can be made continuous at x=0, is

Question

The value of f(0), such that f(x)=(1/x2)(1- cos ( sin x)) can be made continuous at x=0, is (A) (1/2) (B) 2 (C) (1/4) (D) 4

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 0 (1- cos ( sin x)/x2)= displaystyle lim x arrow 0 (2 sin 2(( sin x/2))/x2) displaystyle lim x arrow 0 2 ⋅(( sin (( sin x/2))/(( sin x)2)))2 ⋅(( sin x/2 x))2=(1/2) Thus, for continuity at x=0, f(.0.)=(1/2)

Q. The value of $f (0)$ , such that $f (x) = \frac{1}{x ^{2}} (1 - cos (sin x))$ can be made continuous at $x = 0,$ is

$\frac{1}{2}$

$2$

$\frac{1}{4}$

$4$

$x \to 0 lim \frac{1 - cos ( sin x )}{x ^{2}} = x \to 0 lim \frac{2 sin ^{2} ( \frac{s i n x}{2} )}{x ^{2}} x \to 0 lim 2 \cdot (\frac{sin ( \frac{s i n x}{2} )}{( \frac{s i n x}{2} )})^{2} \cdot (\frac{sin x}{2 x})^{2} = \frac{1}{2}$
Thus, for continuity at $x = 0,$
$f (0) = \frac{1}{2}$

Q. The value of f(0), such that f(x)=x21​(1−cos(sinx)) can be made continuous at x=0, is

21​

2

41​

4

x→0lim​x21−cos(sinx)​=x→0lim​x22sin2(2sinx​)​x→0lim​2⋅((2sinx​)sin(2sinx​)​)2⋅(2xsinx​)2=21​ Thus, for continuity at x=0, f(0)=21​

Q. The value of $f (0)$ , such that $f (x) = \frac{1}{x ^{2}} (1 - cos (sin x))$ can be made continuous at $x = 0,$ is

$\frac{1}{2}$

$2$

$\frac{1}{4}$

$4$

$x \to 0 lim \frac{1 - cos ( sin x )}{x ^{2}} = x \to 0 lim \frac{2 sin ^{2} ( \frac{s i n x}{2} )}{x ^{2}} x \to 0 lim 2 \cdot (\frac{sin ( \frac{s i n x}{2} )}{( \frac{s i n x}{2} )})^{2} \cdot (\frac{sin x}{2 x})^{2} = \frac{1}{2}$
Thus, for continuity at $x = 0,$
$f (0) = \frac{1}{2}$