A question of formality regarding limits Let $f$ be a differentiable function over $\mathbb{R}$, with its derivative being a continuous function. Let there be a function $g$ s.t. $\lim_{x \to 0} g(x) = 0$.

Now I'm required to show that $f'(0) = \lim_{x \to 0} \frac{f(x+g(x)) - f(g(x))}{x}$

The solution seems almost obvious from the definition yet I find myself dealing with two variables and two limits, my question is how would I formally write the transitions leading to this equality, using the definition of the derivative at $x = 0$?

Thanks a million!
from the principal definition of a derivative we know that :

$$f'(t) = \lim_{x \to 0} \frac{f(t + x) - f(t)}{ x}$$

for $f '(0)$ instead of substituting $0$ in the definition we use $g(x)$ since its limit is zero as $x \to 0$.

we thus get:

$$f '(0) = \lim_{x \to 0} \frac{f(g(x) + x) - f(g(x))}{x}$$
from the principal definition of a derivative we know that :

$$f'(t) = \lim_{x \to 0} \frac{f(t + x) - f(t)}{ x}$$

for $f '(0)$ instead of substituting $0$ in the definition we use $g(x)$ since its limit is zero as $x \to 0$.

we thus get:

$$f '(0) = \lim_{x \to 0} \frac{f(g(x) + x) - f(g(x))}{x}$$