If $f(n) = o(g(n))$ then $f(n) + g(n) = \theta(g(n))$?

Question

I'm fairly certain that in this case $f(n) + g(n) =\Theta(g(n))$ is true, but I'm having trouble to proof this. Can someone please put me in the right direction?

Answer 1

$$f(n) \in o(g(n))\Longleftrightarrow \lim_{n\to\infty} \frac{f(n)}{g(n)} = 0$$ $$f(n) + g(n) \in \Theta(g(n)) \Longleftrightarrow \exists c:\lim_{n\to\infty} \frac{f(n) + g(n)}{g(n)} = c$$

Can you prove $\exists c:\lim_{n\to\infty} \frac{f(n) + g(n)}{g(n)} = c$ given $\lim_{n\to\infty} \frac{f(n)}{g(n)} = 0$?