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- Applications of differentiation

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- Lesson: 2a2:00
- Lesson: 2b2:15

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".

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 →$c$} $\frac{f'(x)}{g'(x)}$

or

Then according to l’Hôpital’s Rule: $\lim$

- 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 →$\infty$}$\frac{\ln x}{x-1}$

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