Q38 / 75 · JEE Main 2025 (7 Apr Shift 2) · Mathematics
If the function $f(x)=\frac{\tan (\tan x)-\sin (\sin x)}{\tan x-\sin x}$ is continuous at $\mathrm{x}=0$, then $f(0)$ is equal to ________
Correct answer: 2
Solution
$\lim _{x \rightarrow 0} \frac{\frac{\tan (\tan x)-\tan x}{\tan ^3 x} \frac{\tan ^3 x}{x^3}+\frac{\tan x-\sin x}{x^3}+\frac{\sin x-\sin (\sin x)}{\sin ^3 x} \frac{\sin ^3 x}{x^3}}{\frac{\tan x-\sin x}{x^3}}$
$=\frac{\frac{1}{3}+\frac{1}{2}+\frac{1}{6}}{\frac{1}{2}}=2$
$=\frac{\frac{1}{3}+\frac{1}{2}+\frac{1}{6}}{\frac{1}{2}}=2$
Hence, the correct answer is 2.