Question

I have a question dealing with limits. (conceptual)

(see next post)

Answer 1

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 \[(\lim_{x \rightarrow 0} {\tan^x \over x}) ^8 = (1)^8\]

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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.

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

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