\[ \lim _{x\to a}\left (f(x) \cdot g(x)\right ) = \left (\lim _{x\to a} f(x) \right )\cdot \left (\lim _{x\to a} g(x)\right ). \]
Here we see a dialogue where students discuss combining limits with arithmetic.
Check out this dialogue between two calculus students (based on a true story):
- 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 \[ \lim _{x\to a} \frac {f(x)}{g(x)} = \frac {\lim _{x\to a} f(x)}{\lim _{x\to a} g(x)}. \]
Give an argument (similar to the one above) supporting the idea that
For the next problems, suppose \(L\) is a number near \(1\) and that \(M\) is a number near \(0\).
Using the context above,
\[ \frac {\text {large}}{\text {small}} = ? \]
“large” “small” impossible to say
Using the context above,
2025-01-06 19:49:30 \[ \frac {\text {small}}{\text {small}} = ? \]
“large” “small” impossible to say