Devyn

Riley, I’ve been thinking about limits.

Riley

So awesome!

Devyn

Think about $$\lim _{x\to a} \left (f(x) + g(x)\right ).$$ This is the number that $f(x) + g(x)$ gets nearer and nearer to, as $x$ gets nearer and nearer to $a$.

Riley

You know it!

Devyn

So I think it is the same as $$\lim _{x\to a} f(x) + \lim _{x\to a}g(x).$$

Riley

Yeah, that does make sense, since when you add two numbers, say $$(\text {a number near $6$}) + (\text {a number near $7$})$$ you get $$(\text {a number near $13$})$$

Riley

Right! And I think the same reasoning will work for multiplication! So we should be able to say $$\lim _{x\to a}\left (f(x) \cdot g(x)\right ) = \left (\lim _{x\to a} f(x) \right )\cdot \left (\lim _{x\to a} g(x)\right ).$$

Devyn

Yes, I think that’s right! But what about division? Can we use similar reasoning to conclude $$\lim _{x\to a} \frac {f(x)}{g(x)} = \frac {\lim _{x\to a} f(x)}{\lim _{x\to a} g(x)}.$$

For the next problems, suppose $L$ is a number near $1$ and that $M$ is a number near $0$.