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I have a question dealing with limits. (conceptual) (see next post)

Mathematics
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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 \[(\lim_{x \rightarrow 0} {\tan^x \over x}) ^8 = (1)^8\]
I did the last step with lhospoitals' rule, by the way. done in one step.
I'm wondering if it's ok to "bring the limit" inside the brackets, ignoring the ^8 exponential

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your first step is a bit odd, and im not sure you can do exactly that...
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.
yeah, I know the algebraic manipulations are OK, but is it ok to disregard the ^8 and bring the limit inside the brackets?
Seems okay to me. The resulting limit is the correct one, at least. This is interesting, though. Never thought about it :-) Kudos, mate.
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
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

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