exp(a) goes to 0 and exp(a)exp(b) goes to infinity. What's a and b?

Question

Suppose it is given that

$e^{(a+b)} = e^a e^b \rightarrow \infty$

and that

$e^a \rightarrow 0$.

Is it possible to state mathematically how the values for $a$ and $b$ that fulfill this look like? Intuitively, $a$ should have to go to $-\infty$ while $b$ must go to $+\infty$, but ''faster'' if that is even a mathematical concept in that context.

Answer 1

Let $a$ and $b$ be functions of x.

Let $a(x)=-x$ and $b(x)=2x$

Then it is easy to see that $$e^a\rightarrow 0\ \ \ as\ \ \ x\rightarrow \infty\ \ \ or \ \ \ a\rightarrow -\infty$$and$$e^b\rightarrow \infty\ \ \ as\ \ \ x\rightarrow \infty\ \ \ or \ \ \ b\rightarrow \infty$$and$$e^{a+b}\rightarrow \infty\ \ \ as\ \ \ x\rightarrow \infty\ \ \ or \ \ \ b\rightarrow \infty\ \ \ \ \&\ \ \ a\rightarrow -\infty$$