TITLE: $\lim (a + b)\;$ when $\;\lim(b)\;$ does not exist?
QUESTION [5 upvotes]: Suppose $a$ and $b$ are functions of $x$. Is it guaranteed that
$$
\lim_{x \to +\infty} a + b\text{ does not exist}
$$
when
$$
\lim_{x \to +\infty} a = c\quad\text{and}\quad
\lim_{x \to +\infty} b\text{ does not exist ?}
$$

REPLY [5 votes]: Just to complete joriki's answer:
His answer is right only if $c$ is a number, which seems to be the case from the way you wrote the question. 
Anyhow if you deal also with infinite limits,  it is possible that $\lim_{x \to \infty}b$ does not exist and $\lim_{x \to \infty} a+b$ exists.
Just take $a=x-\sin(x)$ and $b=\sin(x)$.