Please solve this limit.
Its answer is [(2ab)/(a^2-b^2)].

Question

Please solve this limit.
Its answer is [(2ab)/(a^2-b^2)].

maheshmeghwal9 · Answer

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zarkon · Answer

'Its answer is [(2ab)/(a^2-b^2)].'

Really?

anonymous · Answer

wow.... where did the a and b come from?

maheshmeghwal9 · Answer

oops sorry! @zarkon & @dpaInc the answer is "1".

zarkon · Answer

@nbouscal
then what would the answer be if it was $$\lim_{x\to 0}x^{x^x}$$

anonymous · Answer

That limit would still be one. Semi-rigorously: $$
\lim_{x\to0}x^{x^x}=\lim_{x\to0}(x^x)^x\ \lim_{x\to0}x^x=\lim_{x\to0}e^{x\ln x}\
\lim_{x\to0}x\ln x=0 \implies \lim_{x\to0}e^{x\ln x}=1\
\lim_{x\to0}x^{x^x}=\lim_{x\to0}1^x=1^0=1
$$If we wanted to be fully rigorous, we could probably find a delta pretty easily, but I don't want to mess with that.

zarkon · Answer

the limit for the problem I proposed is zero

anonymous · Answer

Right, you can use the same method I used to show that it becomes $0^1$ rather than $1^0$. Shows how much I know about annoying limits. :P