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## Homework Statement

Prove that if ##\lim_{x\to a}f(x)=A\neq 0## then ##\lim_{x\to a}1/f(x)=1/A##

## Homework Equations

This is a proof, so it's just an epsilon delta proof. I know the solution. I am asking about a thought process.

## The Attempt at a Solution

Pick a ##\delta_1## small enough such that if ##\left|x-a\right|<\delta_1## then ##f(x)>A/2##

Then...

##\left|\frac{1}{f(x)}-\frac{1}{A}\right|=\left|\frac{f(x)-A}{f(x)A}\right|<\frac{2}{A^2}\left|f(x)-A\right|##

Now that it is in the form of ##\left|f(x)-A\right|## Now choose an ##\epsilon=\frac{A^2}{2}\epsilon## such that if ##\left|x-a\right|<\delta## then ##\left|f(x)-A\right|<\frac{A^2}{2}\epsilon## then ##\left|\frac{1}{f(x)}-\frac{1}{A}\right|=\frac{2}{A^2}\frac{A^2}{2} \epsilon = \epsilon## This completes the proof.

I understand why this proof works, but I do not understand the thought process to getting to this proof. There is some sort of creativity or accuracy in guessing an appropriate value for the choice of epsilon that completely eludes me. I've struggled over this for weeks and I have not had much leeway at all. If you were to begin this proof, how would you go about approaching the problem to choose the appropriate values for epsilon, and the first limit? I am specifically focusing on why A/2 was chosen (but I understand why A/3 or 3A/4 would also work, and why that would alter the subsequent epsilon). If I were just starting this problem out, what procedure would I pursue in my head to organize this and come up with this solution as a proof?