Proving $\frac{1}{x} \to \frac{1}{3}$ as $x \to 3$

[ciubba]

Homework Statement

Prove lim_{x->3}\frac{1}{x}=\frac{1}{3}

Homework Equations

Epsilon/delta definition

The Attempt at a Solution

|\frac{1}{x}-\frac{1}{3}|<\epsilon \; \; \mbox{when} \; \; |x-3|<\delta
I expanded the left to get
-\epsilon+\frac{1}{3}<\frac{1}{x}<\epsilon+\frac{1}{3}

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

-\delta+3<x<\delta+3

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

2&lt;x&lt;4

|x|&lt;1

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?

 

[O_o]

Fix
\epsilon &gt; 0 and \delta = \min\{1, 6\epsilon\}
Assume 0 &lt; |x- 3| &lt; \delta
then 0 &lt;|x - 3| &lt; \delta \leq 1 \implies 2 &lt; x &lt; 4 \implies \frac{1}{x} &lt; \frac{1}{2}
Now \left| \frac{1}{x} - \frac{1}{3}\right| = \frac{1}{|3x|} | 3 - x | =<br /> \frac{1}{|3x|} | x - 3| &lt; \frac{1}{3}\frac{1}{2} \delta \leq \frac{1}{6} 6\epsilon = \epsilon

 

[ciubba]

I don't understand how you transitioned from 2<x<4 to 1/x<1/2

I also don't see from where you get the 1/3 in <1/3*1/2

 

[O_o]

If 2 &lt; x &lt; 4 then that's the same as saying 2 &lt; x and x &lt; 4 You can manipulate inequalities the same way you do equalities so this means \frac{1}{x} &lt; \frac{1}{2} and \frac{1}{4} &lt; \frac{1}{x} We only need the 1/2 term since we're looking for an upper bound on 1/x. The 1/3 comes from the fact that we have \frac{1}{3} \frac{1}{|x|} so if 1/x < 1/2 then \frac{1}{3}\frac{1}{x} &lt; \frac{1}{3} \frac{1}{2}

 

[ciubba]

Ah, I didn't know that you could do that with inequalities.

I still cannot find <br /> \frac{1}{3} \frac{1}{|x|} anywhere in the proof. Can you point out on which step that relationship can be found?

Edit: NVM, found it. Thanks!