Allowed and not allowed operation with limits

Question

I have been reviewing some properties of limits, and I had the following silly doubt:

If

$$\displaystyle \lim_{x \rightarrow 0} \displaystyle\frac{\ln(1+ax)}{bx}=\frac{a}{b},$$

then is it allowed to rewrite the above equation as

$$\displaystyle\lim_{x \rightarrow 0} \displaystyle\frac{\ln(1+ax)}{bx}\left(\frac{b}{a}\right)=1 ?$$

In order words, after one evaluates the limit, is it possible to perform algebraic manipulations like the one presented above?

Answer 1

$$\lim_{x \to 0} cf(x) = c \lim_{x \to 0} f(x)$$ assuming the limits exist. Your particular example is $c=b/a$ and $f(x) = \frac{\ln(1+ax)}{bx}$.

Answer 2

Yes, since, after having proved that$$\lim_{x\to0}\frac{\log(1+ax)}{bx}=\frac ab,$$then $\lim_{x\to0}\frac{\log(1+ax)}{bx}$ is simply the number $\frac ab$. And $\frac ab\times\frac ba=1$.

Answer 3

No, you can't.

Unless you know that $a\ne 0$. (We must already be given that $b\ne 0$)