What is the linear approximation of $f(x) = \frac1x$ when $x \approx 0$?

Question

I did not find an example when the denominator $x$ approximates to $0$.

$f(0) + f'(0)x$ does not work because $f(0)$ would be $+\infty$.

Answer 1

Well, $f(x) = \frac{1}{x}$ is not even defined at $x=0$ which obviously means it can't be differentiated and in particular we cannnot find a linear approximation on that point.

Answer 2

There is no answer to your question as the limit does not exist. This can be proven by-

1. first take the left hand side limit. This is equal to negative infinity in this case
2. Then take the right hand side limit, which equals positive infinity
3. Since, left hand side limit is not equal to right hand side limit, the limit does not exist.

I hope you got your answer.