l'Hospital's rule

l'Hospital's rule

Remember that one tricky Limits section that required intense algebraic manipulation to avoid getting 0/0 or infinity/infinity limits? We will now revisit it again, but with the knowledge of derivatives. In this section, we will learn how derivatives enable us to efficiently evaluate the limits of a function using the "L'Hospital's rule".

Lessons

Note *l’Hôpital’s Rule applies to 2 types of indeterminate forms:

type $\frac{0}{0}$ (that is, $\lim$x →$c$ $f(x)=0$ and $\lim$x →$c$$g(x)=0$)
or
type $\frac{\infty}{\infty}$ (that is, $\lim$x →$c$ $f(x)=\pm \infty$ and $\lim$x →$c$$g(x)=\pm \infty$)

Then according to l’Hôpital’s Rule: $\lim$x →$c$ $\frac{f(x)}{g(x)}=$ $\lim$x →$c$ $\frac{f'(x)}{g'(x)}$
• 1.
Evaluating the limit of the form:
$\lim$x →$c$ $\frac{f(x)}{g(x)}$

• 2.
Evaluating the limit.
Find:
a)
$\lim$x →$1$ $\frac{\ln x}{x-1}$

b)
$\lim$x →$\infty$ $\frac{\ln x}{x-1}$