I have been trying some questions on the convergence of a sequence of functions and was wondering about an intermediate step in which we have $\epsilon\gt0$ and an $x$ such that $0<x<1$ and it says that if $\epsilon$ < 1 then $\frac{\log(1/\epsilon)}{\log(1/x)}$ will tend to infinity as $x$ tends to 1 from the left-hand side.
I just wanted to know where will it tend to if $\epsilon\geq 1$ as $x$ tends to $1$ from the left? (for $1$, it seems to tend to $0$, but what for $\epsilon\gt1$?)