Nested Indeterminate Forms

Suppose $f$ and $g$ are differentiable functions such that

$g'(x) \neq 0$ on an open interval $I$ containing $0$ ;
$\lim_{x \to 0} f(x) = 0$ and $\lim_{x \to 0} g(x) = 0$ ;
$\lim_{x \to 0} \frac{f'(x)}{g'(x)} = 0$ .

L'Hôpital's Rule concludes that we can find $\lim_{x \to 0} \frac{f(x)}{g(x)} = \lim_{x \to 0} \frac{f'(x)}{g'(x)} = 0$ .

What can we conclude about how to find $\lim_{x \to 0} \frac{f(\frac{f(x)}{g(x)})}{g(\frac{f(x)}{g(x)})}$ ?

Example question: put $f(x) = \sin(x) - x$ and $g(x) = x$ , find $\lim_{x \to 0} \frac{\sin(\frac{\sin(x) - x}{x}) - (\frac{\sin(x) - x}{x})}{(\frac{\sin(x) - x}{x})} = \lim_{x \to 0} \frac{\sin(\frac{\sin(x)}{x} - 1) - (\frac{\sin(x)}{x} - 1)}{(\frac{\sin(x)}{x} - 1)}$ ?

#Calculus

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