Quotient Limit Law: Find the Value of the Limit

[FS98]

The quotient limit laws says that the limit of a quotient is equal to the quotient of the limits.

If we had a limit as x approaches 0 of 2x/x we can find the value of that limit to be 2 by canceling out the x’s.

If we split it up we get the limit as x approaches 2 of 2x divided by the limit as x approaches to of x. That would give us 0/0 so wouldn’t the new answer be undefined?

I must be doing something wrong because I think they should be giving the same answer.

 

[mfb]

You omitted an important part of the quotient limit law. The limit for the denominator cannot be zero, otherwise the law is not applicable.
(In addition, both limits have to exist)

 

[Mark44]

That would give us 0/0 so wouldn’t the new answer be undefined?

0/0 isn't a number, but the usual terminology is the indeterminate form $\frac{[0]}{[0]}$, which is just one of several indeterminate forms that can arise from limits.

When you take a limit and arrive at this indeterminate form, it doesn't mean that the limit isn't defined -- all it means is that you can't tell without additional work whether the limit exists.

Here are three examples of the indeterminate form $\frac{[0]}{[0]}$, all with different results.
$\lim_{x \to 0} \frac {2x} x$ -- limit exists and is 2.
$\lim_{x \to 0} \frac {x} {x^2}$ -- limit does not exist.
$\lim_{x \to 0} \frac {x^3} x$ -- limit exists and is 0.