Q. 90 - Calculus

Question

In your own words, prove the special case of L’Hopital’s ˆ

rule that is proved in the reading. Explain each step in

detail.

Step-by-Step Solution

Answer

The special case of L’Hopital’s ˆ rule is proved

Step 1. Given information

To prove the special case of L’Hopital’s ˆ rule 

Step 2. Prove the special case

$$\begin{gathered} \underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\lim }\frac{{\displaystyle \frac{f\left(x\right)-f\left(a\right)}{x-a}}·\left(x-a\right)+f\left(a\right)}{{\displaystyle \frac{g\left(x\right)-g\left(a\right)}{x-a}}·\left(x-a\right)+g\left(a\right)} \\ =\underset{x\to a}{\lim }\frac{{\displaystyle \frac{f\left(x\right)-f\left(a\right)}{x-a}}.\left(x-a\right)+0}{{\displaystyle \frac{g\left(x\right)-g\left(a\right)}{x-a}}·\left(x-a\right)+0} \\ =\underset{x\to a}{\lim }\frac{\lambda -\upsilon }{{\displaystyle \frac{g\left(x\right)-g\left(a\right)}{x-a}}} \\ =\frac{f\text{'}\left(a\right)}{g\text{'}\left(a\right)} \\ =\frac{\underset{x\to a}{\lim }f\text{'}\left(x\right)}{\underset{x\to a}{\lim }g\text{'}\left(x\right)} \end{gathered}$$

Step 3. The solution

The special case of L’Hopital’s ˆ rule is proved