This may seem like a simple question, but I feel as if it is wrong but I am unsure why. Is it possible to evaluate a limit in two stages for example: say you know that $x(1- a)\rightarrow b$ as $x\rightarrow \infty$, would I be correct in saying
$(1 - \frac{x(1 - a)}{x})^x \rightarrow (1 - \frac{b}{x})^x \rightarrow \exp(-b)$ as $x\rightarrow\infty$
Thank you for any help.
No, it is not correct in general to do a "2-stage-limit" as you suggest. Otherwise, you could argue that for instance $$\left(1+\frac{1}{x}\right)^x \xrightarrow[x\to\infty]{} 1$$ "since" $1+\frac{1}{x} \xrightarrow[x\to\infty]{} 1$ and $1^x \xrightarrow[x\to\infty]{} 1$.
it might still yield the right result in some cases, but the derivation cannot be argued that way -- you need to get your hands a bit dirtier.