Math Help - prove or disprove... involving sided limits

1. prove or disprove... involving sided limits

How can I prove the following:
$\lim_{x\to 0}f(x)=L$ implies $\lim_{x\to \infty}f(\frac{1}{x})=L$
For that matter, how would I disprove this:
$\lim_{x\to \infty}f(\frac{1}{x})=L$ implies $\lim_{x\to 0}f(x)=L$

2. Originally Posted by dannyboycurtis
How can I prove the following:
$\lim_{x\to 0}f(x)=L$ implies $\lim_{x\to \infty}f(\frac{1}{x})=L$
For that matter, how would I disprove this:
$\lim_{x\to \infty}f(\frac{1}{x})=L$ implies $\lim_{x\to 0}f(x)=L$
$\forall~\epsilon>0$, $\exists~\delta>0$ such that $|y-0|=|y|<\delta \implies |f(y)-L|<\epsilon$

Replace $y$ with $\frac{1}{x}$ and we have $|1/x|<\delta \implies |f(1/x)-L|<\epsilon$.

So $\forall~\epsilon>0$, $\exists~\delta>0$ such that $x>1/\delta \implies |f(1/x)-L|$.

This means $\lim_{x\to\infty}f(1/x)=L$.

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I cannot think of a counterexample for the second one. I'm sure someone else will, though.