I have to show $\displaystyle \lim_{x\to 1^+}\frac{x}{x-1}=+\infty$
I have no idea how to begin... any help?
$\displaystyle \frac{x}{x-1}=1+\frac{1}{x-1}$ So you want to prove that $\displaystyle \forall N$, $\displaystyle \exists~\delta>0$ such that $\displaystyle |x-1|<\delta$ implies $\displaystyle 1+\frac{1}{x-1}>N$.
So given an $\displaystyle N$, let $\displaystyle \delta=\frac{1}{N-1}$.