Q:

An Inquiry About the Sum of Limits

The limit law $\lim_{x\to c}[f(x)+g(x)]=\lim_{x\to c}f(x)+\lim_{x\to c}g(x)$ is true given that $\lim_{x\to c}f(x)$ and $\lim_{x\to c}g(x)$ both exist. I also know that $\lim_{x\to c}[f(x)+g(x)]$ can exist even if both $\lim_{x\to c}f(x)$ and $\lim_{x\to c}g(x)$ don't (i.e. if $g=-f$). I was wondering whether  $\lim_{x\to c}[f(x)+g(x)]$ can exist if the limit of one function exists while the limit of the other doesn't, and if not, how I would go about to prove this.
This is for a proof in which I want to make the statement "since I know that $\lim_{x\to c}[f(x)+g(x)]$, either  $\lim_{x\to c}f(x)$ and $\lim_{x\to c}g(x)$ both exist or both don't exist. Since we know $\lim_{x\to c}f(x)$ exists, we therefore know that $\lim_{x\to c}g(x)$ exists."

A:

If both limits$$\lim_{x\to c}\bigl(f(x)+g(x)\bigr)\text{ and }\lim_{x\to c}f(x)$$exist, then the limit $\lim_{x\to c}g(x)$ must exist too, since $g(x)=\bigl(f(x)+g(x)\bigr)-f(x)$.

