$\lim (a + b)\;$ when $\;\lim(b)\;$ does not exist? Suppose $a$ and $b$ are functions of $x$. Is it guaranteed that
$$
\lim_{x \to +\infty} a + b\text{ does not exist}
$$
when
$$
\lim_{x \to +\infty} a = c\quad\text{and}\quad
\lim_{x \to +\infty} b\text{ does not exist ?}
$$

Suppose, to get a contradiction, that our limit exists.  That is, suppose $$\lim_{x\rightarrow \infty} a(x)+b(x)=d$$ exists.  Then since $$\lim_{x\rightarrow \infty} -a(x)=-c,$$ and as limits are additive, we conclude that $$\lim_{x\rightarrow \infty} a(x)+b(x)-a(x)=d-c$$ which means $$\lim_{x\rightarrow \infty} b(x)=d-c.$$  But this is impossible since we had that $b(x)$ did not tend to a limit.
Hope that helps,
