# Math Help - Real Analysis Proof

1. ## Real Analysis Proof

How would you prove that (1/x^n)-->0 as x-->infinity and (1/x^n)-->0 as x-->-infinity?

2. Originally Posted by zebra2147
How would you prove that (1/x^n)-->0 as x-->infinity and (1/x^n)-->0 as x-->-infinity?
Show that for all $\varepsilon>0$ there exists a $X_{\varepsilon}$ such that:for all $x>X_{\varepsilon}$ we have $|x^{-n}-0|<\varepsilon$

In this case choose $X_{\varepsilon}=\max\left(1,\frac{1}{\varepsilon}\ right)$, then for $x>X_{\varepsilon}$:

$|x^{-n}-0|=|x^{-n}|<\frac{1}{x}<\frac{1}{X_{\varepsilon}}\le \varepsilon$

CB

3. Question: I understand what you said besides how you chose ?

4. Originally Posted by zebra2147
Question: I understand what you said besides how you chose ?
You guess, try a few things, find something that works (there is no unique choice that works, there are plenty of them).

CB