Equal or not?

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

Give an argument (similar to the one above) supporting the idea that $\displaystyle \lim _{x\to a}\left (f(x) \cdot g(x)\right ) = \left (\displaystyle \lim _{x\to a} f(x) \right )\cdot \left (\displaystyle \lim _{x\to a} g(x)\right ).$

What can we say about the following expressions? $\frac {\text {large}}{\text {small}} = ?$

‘‘large’’ ‘‘small’’ impossible to say

$\frac {\text {small}}{\text {large}} = ?$

‘‘large’’ ‘‘small’’ impossible to say

Using the context above, $\frac {\text {small}}{\text {small}} = ?$

‘‘large’’ ‘‘small’’ impossible to say

The problem with dividing two small quantities is that you need to know whether one is smaller than the other in order to make any definitive conclusions. For example, both $0.0001$ and $0.00000000001$ are small, but $0.00000000001$ is much smaller than $0.0001$ , so $\frac {0.0001}{0.00000000001} = 10000000$ , which is pretty large! On the other hand, $\frac {0.00000000001}{0.0001} = 0.0000001$ , which is pretty small. In both cases, we’re considering a $\frac {\text {small}}{\text {small}}$ situation, but the result could be either large or small depending on which number is in the numerator and what is in the denominator.

Using the context above, $\frac {\text {large}}{\text {large}} = ?$

‘‘large’’ ‘‘small’’ impossible to say

Just like with the $\frac {\text {small}}{\text {small}}$ case, if you don’t know how the large quantities compare to one another, your fraction could be either very large or very small.