Answer
$\lim\limits_{x\to c} g(x)$ does not exist.
Work Step by Step
The limit property suggests that $\lim\limits_{x \to c}[f(x)+g(x)]=\lim\limits_{x\to c} f(x)+\lim\limits_{x\to c} g(x)$
Let us assume that the limit $\lim\limits_{x\to c} f(x)$ exists.
Consider $\lim\limits_{x\to c} g(x)=P$ and so, $\lim\limits_{x \to c}[f(x)+g(x)]=Q$
Now, $Q=\lim\limits_{x\to c} f(x)+\lim\limits_{x\to c} g(x) \implies Q=\lim\limits_{x\to c} f(x)+P$
This implies that $\lim\limits_{x\to c} f(x)=Q-P$, that is, the limit exists.
Hence, the limit $\lim\limits_{x\to c} g(x)$ does not exist.
