AP Calculus AB / Unit 4: Contextual Applications of Differentiation
4.7 Using L'Hôpital's Rule for Determining Limits of Indeterminate Forms
Notes
If a limit gives $\displaystyle \frac{0}{0}$ or $\displaystyle \frac{\infty}{\infty}$, then $$\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}$$ when the right side exists.
- Differentiate the top and bottom separately (not the quotient rule).
- Confirm the indeterminate form before using it.
Practice
$\displaystyle \lim_{x\to0}\frac{e^x-1}{x}=$
$\displaystyle \frac{e^x}{1}\to1$.
$\displaystyle \lim_{x\to\infty}\frac{\ln x}{x}=$
$\displaystyle \frac{1/x}{1}\to0$.
Can L'Hôpital's Rule be used on $\displaystyle \lim_{x\to1}\frac{x+1}{x}$?
Substitution gives 2 directly.
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