Multi-variable Limits

Question

If we look at the following expression,

$\lim_{x\rightarrow \infty} \frac{e^{tx}}{x^{2}}$; $t>0$

then it should diverge to $\infty$ because $e^{tx}$ diverges to $\infty$ faster than $x^{2}$ when $t>0$. However, if we go closer and closer to $0$, then this differential in rates would lessen, and finally at $t=0$, the limit actually is $0$. My question is regarding the behaviour of the expression when $t\rightarrow 0$. Without knowing how $t$ and $x$ are related, can we say anything meaningful about the expression-

$\lim_{x\rightarrow \infty; t\rightarrow 0} \frac{e^{tx}}{x^{2}}$

Answer 1

This limit makes sense, but it does not exist. Indeed, if $t=\frac1x$, then it becomes $$\lim_{x\to+\infty}\frac e{x^2}=0,$$but if $t=\frac1{\sqrt x}$, then it becomes$$\lim_{x\to+\infty}\frac{e^{\sqrt x}}{x^2}=+\infty.$$