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

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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.

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

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