## ggrree 3 years ago I have a question dealing with limits. (conceptual) (see next post)

1. ggrree

say you had: $\lim_{x \rightarrow 0} {\tan^8x \over x^8}$ would it be correct to say this?$\lim_{x \rightarrow 0} ({\tan^x \over x})^8$ and then $(\lim_{x \rightarrow 0} {\tan^x \over x}) ^8 = (1)^8$

2. ggrree

I did the last step with lhospoitals' rule, by the way. done in one step.

3. ggrree

I'm wondering if it's ok to "bring the limit" inside the brackets, ignoring the ^8 exponential

4. agreene

your first step is a bit odd, and im not sure you can do exactly that...

5. agreene

after looking at this again--i thought they were different exponents... $\large \frac{\tan^8 x}{x^8}=(\frac{\tan x}{x})^8$ is true.

6. ggrree

yeah, I know the algebraic manipulations are OK, but is it ok to disregard the ^8 and bring the limit inside the brackets?

7. bmp

Seems okay to me. The resulting limit is the correct one, at least. This is interesting, though. Never thought about it :-) Kudos, mate.

8. bmp

I wonder if that's the same as taking repetitive L'Hopital rules? I mean, since tanx/x will be 1 as x approaches 0

9. gordonj005

i seem to remember this from calc: it's ok to bring the limit inside a function if the function is one-to-one on the interval in question, this is to preserve inequalities in the epsilon-delta definition of the limit