If the function is continuous at x = 0, then f(0) is equal to ________. [2025]

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Solution

Q.
If the function $f (x) = \frac{\tan (\tan x) - \sin (\sin x)}{\tan x - \sin x}$ is continuous at x = 0, then f(0) is equal to ________. [2025]

Ans.
(2)

$f (0) = \lim_{x \to 0} \frac{\tan (\tan x) - \sin (\sin x) - \tan x + \tan x - \sin x + \sin x}{\tan x - \sin x}$

$= \lim_{x \to 0} \frac{\frac{\tan (\tan x) - \tan x}{\tan^{3} x} \times \frac{\tan^{3} x}{x^{3}} + \frac{\tan x - \sin x}{x^{3}} + \frac{\sin x - \sin (\sin x)}{\sin^{3} x} \times \frac{\sin^{3} x}{x^{3}}}{\frac{\tan x - \sin x}{x^{3}}}$

$= 1 + \frac{(\frac{1}{3} + \frac{1}{6})}{(\frac{1}{3} + \frac{1}{6})} = 2$ . [From L'Hospital's Rule]

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