I have two arbitrary functions f(x) and g(x), for which $\lim_{x\rightarrow \infty} f(x) = \infty$ and $\lim_{x\rightarrow \infty} g(x)=\infty$.
1.) Are there any general conditions that would ensure that $\lim_{x \rightarrow \infty}\frac{e^{f(x)}}{e^{g(x)}}=0$ (or $\lim_{x \rightarrow \infty}\frac{e^{g(x)}}{e^{f(x)}}=0$)?
2.) I suppose this is the case when g(x) approaches infinity "faster" than f(x)? But is that just tautology? If so, what could then ensure such desired different rates of convergences to infinity.
3.) As long as f(x) and g(x) are different functions, surely they will have different rates of convergence? Thus wouldn't it hold then that always either $\frac{e^f(x)}{e^g(x)}$ or $\frac{e^g(x)}{e^f(x)}$ approaches 0?
4.) I suspect my last statement isn't true as you could maybe find functions that both approach infinity yet the ratio discussed here reaches a finite limit. My question really is if there are any general conditions to ensure that this doesn't happen (i.e. ensure limit is 0 or $\infty$ rather than finite).
Best Answer
You can show for real-valued functions \begin{align} a) & \lim_{x\rightarrow\infty} \frac{e^{f(x)}}{e^{g(x)}} = \infty \iff \lim_{x\rightarrow\infty} (f(x)-g(x)) = \infty\\ b) & \lim_{x\rightarrow\infty} \frac{e^{f(x)}}{e^{g(x)}} = 0 \iff \lim_{x\rightarrow\infty} (f(x)-g(x)) = -\infty \end{align}