You must have misunderstood what the book says.

According to the algebraic limit theorem, $\lim_{x\to a}[f(x)+g(x)]=\lim_{x\to a}f(x)+ \lim_{x\to a}g(x),$

provided the limits on the right side exist.

It is possible for $\lim_{x\to a} [f(x)+g(x) ]$ to exist when $\lim_{x\to a}f(x)$ and $ \lim_{x\to a}g(x)$ do not,

but there are plenty of examples (such as the one you gave in the question)

where $\lim_{x\to a}f(x)$ and $ \lim_{x\to a}g(x)$ do not exist and $\lim_{x\to a} f(x)+g(x)$ does not either.

Perhaps the book was trying to say that you cannot have the situation where

$\lim_{x\to a} [f(x)+g(x) ]$ and $\lim_{x\to a}f(x)$ exist but $ \lim_{x\to a}g(x)$ does not.