I seem to be having problems understanding the epsilon-N definition of limits when the function takes multiple variables.
For example, consider the limit $\lim_{(x,y) \rightarrow (\infty, \infty)} xe^{-y}$, which has come up in my stats homework. My hunch is that this limit should converge to $0$, as this yields the correct answer and the graph seems to "flatten out" in general when looking far away in the first quadrant.
Yet, I can neither confirm nor disprove this guess since I cannot find the definition of limits of multivariable functions at infinity. The only definition I could find are those at finite points, in which case a direct generalization of $\epsilon-\delta$ definition for single variable functions could be applied.
Could somebody please explain the rigorous definition of limits at infinity? Also, if possible, could you confirm or disprove my guess about $\lim_{(x,y) \rightarrow (\infty, \infty)} xe^{-y}$?
Thanks very much.
Best Answer
In this case, the limit is not well-defined. Specifically, it depends on the path you take to get to $(\infty, \infty)$. For example, if you fix $x$ and take $y$ to $\infty$, you will see that the function goes to zero everywhere. If you then take $x$ to infinity, well zero stays zero. If you do it in the opposite order (fix $y$ and take $x$ to $\infty$, then take $y$ to $\infty$), you will get that the function blows up.
In general, multivariate functions -- even nice continuous, smooth ones like $xe^{-y}$ -- will not have good limits as you go to infinity. You would need another property (like uniform convergence) to talk about the limit as you go to $(\infty,\infty)$.