texelaare

2021-10-28

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.
$\underset{x\to 4}{lim}\frac{\frac{1}{x}-\frac{1}{4}}{x-4}$

Sally Cresswell

Given limit:
$\underset{x\to 4}{lim}\left(\frac{\frac{1}{x}-\frac{1}{4}}{x-4}\right)$
Now,
$\underset{x\to 4}{lim}\left(\frac{\frac{1}{x}-\frac{1}{4}}{x-4}\right)=\underset{x\to 4}{lim}\left(\frac{\frac{4-x}{4x}}{x-4}\right)$
$=\underset{x\to 4}{lim}\left(\frac{4-x}{4x\left(x-4\right)}\right)$
$=\underset{x\to 4}{lim}\left(\frac{-\left(x-4\right)}{4x\left(x-4\right)}\right)$
$\underset{x\to 4}{lim}\left(-\frac{1}{4x}\right)$
Plug in the value x=4:
$=-\frac{1}{4.4}$
$=-\frac{1}{16}$
Thus,
$\underset{x\to 4}{lim}\left(\frac{\frac{1}{x}-\frac{1}{4}}{x-4}\right)=-\frac{1}{16}$

Jeffrey Jordon