L'Hôpital's rule

Motivation. Consider the problem of computing the following limit: $$\lim_{x \rightarrow 0}\frac{\sin(x)}{x}.$$ How should you go about it? My favorite answer is to see this as a derivative. Namely, we have $$\begin{align*}\lim_{x \rightarrow 0}\frac{\sin(x)}{x} &= \lim_{x \rightarrow 0}\frac{\sin(x) - \sin(0)}{x - 0} \\ &= \left.\frac{d}{dx}\right|_{0} \sin(x) \\ &= \left. \cos(x)\right|_{x=0} \\ &= 1.\end{align*}$$ But then it might not be so straight forward to figure out the limit like $$\lim_{t \rightarrow 0}\frac{\sin(x)}{\sin(\sin(x^{3}))}.$$ This is why we learn L'Hôpital's rule:

Exercise. Read p.253, especially two boxes outlined in blue: these are L'Hôpital's rules.

Exercise. Compute $$\lim_{x \rightarrow 0}\frac{\sin(x)}{\sin(\sin(x^{3}))}.$$

Exercise (Hard). Compute $$\lim_{x \rightarrow 0+}x^{x}.$$

Exercise. Compute $$\lim_{x \rightarrow \infty}\frac{x^{2} + x + 1}{\ln(x)}.$$

Exercise. Compute $$\lim_{x \rightarrow \infty}\frac{x^{2} + x + 1}{3^{x}}.$$

Exercise. Compute $$\lim_{t \rightarrow \infty}(1 + t)^{t}.$$