Hi! I have to find a counterexample to the following statement:

$\displaystyle \lim_{x\to \infty}f(\frac{1}{x})=L$ implies $\displaystyle \lim_{x\to 0}f(x)=L$

I assert that $\displaystyle f(x)=\left\{\begin{array}{lr}1:&x>0\\-1:&x\leq 0\end{array}\right\}$ is a viable counterexample.

Is this correct?

The limit at 0 does not exist, but the limit of f(x) as x goes to infinity will be 1.

Im just not 100% sure...

Any feedback would be appreciated.