I am trying to calculate $\lim_{(x,y)\rightarrow(0,0)}\frac{e^{1-xy}}{\sqrt{2x^2+3y^2}}$, or show that the limit does not exist.
Thoughts: In the limit $(x,y) \to (0,0)$, it seems that the numerator is tending to a constant value $e$, while the denominator tends to zero, so the required limit might be infinity. However, I am not sure how to obtain a suitable lower bound to show this.
I also tried to show that the limit does not exist, without much luck. Approaching the origin along the $x$-axis would give: $\lim_{x\to 0}\frac{e}{x\sqrt{2}}$, and along the $y$-axis would give $\lim_{y\to 0}\frac{e}{y\sqrt{3}}$. Both these limits tend to infinity. Approaching $(0,0)$ along $y=x$ also suggests the same.
Best Answer
Yes you are right, since numerator tends to $e$ and the denominator to $0^+$ we can easily conclude that the limit is $\infty$, indeed eventually by squeeze theorem
$$\frac{e^{1-xy}}{\sqrt{2x^2+3y^2}}\ge \frac 1{\sqrt{2x^2+3y^2}} \ge \frac 1{\sqrt 3\sqrt{x^2+y^2}}\to \infty$$