쌍곡선함수의 미적분

List the different indeterminate forms described in this section.

T/F: l'Hopital's Rule provides a faster method of computing derivatives.

T/F: l'Hopital's Rule states that $\displaystyle \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f\,'(x)}{g'(x)}.$

Explain what the indeterminate form “$1^\infty$” means.

Fill in the blanks:

The Quotient Rule is applied to $\displaystyle \frac{f(x)}{g(x)}$ when taking _____;

l'Hopital's Rule is applied to $\displaystyle \frac{f(x)}{g(x)}$ when taking certain _____.

Create (but do not evaluate!) a limit that returns “$\infty^0$”.

Create a function $f(x)$ such that $\displaystyle \lim_{x\to 1}f(x)$ returns “$0^0$”.

Create a function $f(x)$ such that $\displaystyle \lim_{x\to \infty}f(x)$ returns “$0\cdot\infty$”.