Proving that if $\lim_{(x,y)\to(a,b)} f(x,y) = \infty$, then $\lim_{(x,y)\to(a,b)} \frac{ln(f(x,y))}{f(x,y)}=0$

Question

How can I properly prove this using definitions? Does the hypothesis allow me to get rid of the denominator in $|\frac{\ln(f(x,y))}{f(x,y)}|$?

Answer 1

Use the fact that $\lim_{t \to \infty} \frac {\ln t} t =0$. If $\epsilon >0$ there exists $M$ such that $|\frac {\ln t} t| <\epsilon$ for $t >M$. Now there exists $\delta >0$ such that $f(x,y) >M$ for $d((x,y),(a,b))<\delta$. Just put these two inequalities together.