I have a homework problem that I'm not sure how to start. I tried Google for similar examples it didn't turn up anything. Could someone tell me the name of the concept to look into? The problem is as follows:

Show by example that $\lim_{x\to c}f(x) + g(x)$ can exist even if both $\lim_{x\to c}f(x)$ and $\lim_{x\to c}g(x)$ do not exist.

## Best Answer

Let $f(x)=\frac{1}{x}$, $g(x)=-\frac{1}{x}$, and $c=0$. We note that clearly

$$\lim_{x\to0}f(x)=\lim_{x\to0}\frac{1}{x} \text{ d.n.e}$$

and that

$$\lim_{x\to0}g(x)=\lim_{x\to0}-\frac{1}{x} \text{ d.n.e}$$

However

$$\lim_{x\to0}f(x)+g(x)=\lim_{x\to0}\frac{1}{x}-\frac{1}{x}=0$$