Let functions f(x) and g(x) be defined in some neighborhood of point a, such that $$\lim\limits_{x\to a}{f(x)}=b$$
And the following limit does not exist:
$$\lim\limits_{x\to a}{g(x)}$$
Choose the wrong assumption:
Might exist:
$$\lim\limits_{x\to a}{(f(x) + g(x))} $$
$$\lim\limits_{x\to a}{\frac{f(x)}{g(x)}} $$
The only rule here I know is when both limits exist, then we can find their sum/product/difference/quotient. But it doesn't seem much of a help.
I know that the wrong assumption is
Might exist: $$\lim\limits_{x\to a}{(f(x) + g(x))} $$
However, why is that so?
If $\lim(f+g)$ exists, then so does $\lim[(f+g)-f]=\lim g$. But you know the latter doesn’t exist, which contradicts the hypothesis.
For the other, consider $a=0$ with $f(x)=1$ and $g(x)=1/x$.