max and min of $f(x,y)=e^{x-y}$

Question

If I consider the restriction to $x$-axis $f(x,0)=e^x \rightarrow +\infty$ for $x \rightarrow +\infty$ so $\sup f(x,y)$ is $+\infty$. If $f(x,y)>0$ can I say $\inf f(x,y)$ is zero?

Answer 1

I don't know why is it that you write if $f(x,y)>0$. If course it is greater than $0$. And, yes, $\inf f(x,y)=0$, since you always have $f(x,y)>0$ and furthermore $\lim_{y\to+\infty}f(x,y)=0$.