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

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

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.
• This is the correct answer. $+1$ – Clayton Apr 19 at 0:05