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

Skilled2021-10-29Added 91 answers

Given limit:

$\underset{x\to 4}{lim}\left(\frac{\frac{1}{x}-\frac{1}{4}}{x-4}\right)$

Now,

$\underset{x\to 4}{lim}\left(\frac{\frac{1}{x}-\frac{1}{4}}{x-4}\right)=\underset{x\to 4}{lim}\left(\frac{\frac{4-x}{4x}}{x-4}\right)$

$=\underset{x\to 4}{lim}\left(\frac{4-x}{4x(x-4)}\right)$

$=\underset{x\to 4}{lim}\left(\frac{-(x-4)}{4x(x-4)}\right)$

$\underset{x\to 4}{lim}(-\frac{1}{4x})$

Plug in the value x=4:

$=-\frac{1}{4.4}$

$=-\frac{1}{16}$

Thus,

$\underset{x\to 4}{lim}\left(\frac{\frac{1}{x}-\frac{1}{4}}{x-4}\right)=-\frac{1}{16}$

Now,

Plug in the value x=4:

Thus,

Jeffrey Jordon

Expert2022-07-03Added 2575 answers