# Nested Indeterminate Forms

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

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

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 $\lim_{x \to 0} \frac{\sin(\frac{\sin(x) - x}{x}) - (\frac{\sin(x) - x}{x})}{(\frac{\sin(x) - x}{x})} = \lim_{x \to 0} \frac{\sin(\frac{\sin(x)}{x} - 1) - (\frac{\sin(x)}{x} - 1)}{(\frac{\sin(x)}{x} - 1)}$ ?

Note by A Former Brilliant Member
5 months, 3 weeks ago

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Sort by:

- 5 months, 3 weeks ago

- 5 months, 3 weeks ago

@Yajat Shamji Do you want the solution of that last limit problem?

- 5 months, 3 weeks ago

Yes! I think the Calculus Geeks can handle this!

If you don't know where Calculus Geeks came from, ask Aruna Yumlembam...

- 5 months, 3 weeks ago

- 5 months, 3 weeks ago