The value of f0, such that fx=1/(x^2)1-cossinx can be made continuous at x=0, is

Correct answer: (A) 1/2. limx→0⁡(1-cossinx)/(x^2)=limx→0⁡(2sin^2⁡sinx)/2x^2 limx→02.sinsinx^2sinx22.sinx^2x2=1/2⁡ Thus, for continuity at x=0, f(0)=1/2

Q8 / 93 · NTA Abhyas (JEE Main) 2020 · Mathematics · Continuity and Differentiability

The value of $f (0)$ , such that $f (x) = \frac{1}{x^{2}} (1 - c o s (s i n x))$ can be made continuous at $x = 0,$ is

Correct answer
A
$\frac{1}{2}$
B
$2$
C
$\frac{1}{4}$
D
$4$

Solution

$\lim_{x \to 0} \frac{1 - c o s (s i n x)}{x^{2}} = \lim_{x \to 0} \frac{2 \sin^{2} (\frac{s i n x}{2})}{x^{2}}$ $\lim_{x \to 0} 2 . {(\frac{\sin (\frac{\sin x}{2})}{(\frac{\sin x}{2})})}^{2} . {(\frac{\sin x}{2 x})}^{2} = \frac{1}{2}$

Thus, for continuity at $x = 0,$

$f (0) = \frac{1}{2}$

Hence, the correct answer is option: 'A'.

The value of f0, such that fx=1x21-cossinx can be made continuous at x=0, is

Solution

More Continuity and Differentiability Questions

The value of $f (0)$ , such that $f (x) = \frac{1}{x^{2}} (1 - c o s (s i n x))$ can be made continuous at $x = 0,$ is