- #1

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## Homework Statement

Prove [tex]lim_{x->3}\frac{1}{x}=\frac{1}{3}[/tex]

## Homework Equations

Epsilon/delta definition

## The Attempt at a Solution

[tex]|\frac{1}{x}-\frac{1}{3}|<\epsilon \; \; \mbox{when} \; \; |x-3|<\delta[/tex]

I expanded the left to get

[tex]-\epsilon+\frac{1}{3}<\frac{1}{x}<\epsilon+\frac{1}{3}[/tex]

I can't turn that into something of the form x-3 without introducing new solutions, so I tried to expand the right side

[tex]-\delta+3<x<\delta+3[/tex]

Which didn't help, so I tried defining ϵ<2/3 so that |x-3|<1, so

[tex]2<x<4 [/tex]

[tex]|x|<1[/tex]

Unfortunately, I don't see any way to turn x into 1/x without inverting the inequality, at which point I'd have > symbols, which doesn't agree with the left side. Any suggestions?