Find the limit. Use L'Hopital's rule if possible. Use a more elementary method if possible. If L'Hopital doesn't apply, explain why.
lim x--0 (x^(x^2))

Question

Find the limit. Use L'Hopital's rule if possible. Use a more elementary method if possible. If L'Hopital doesn't apply, explain why.
lim x-->0 (x^(x^2))

jennychan12 · Answer

$$\lim_{x \rightarrow 0} x^{x^2}$$

jennychan12 · Answer

@agent0smith 
can you help?

anonymous · Answer

don't know how to do with without l'hopital

jennychan12 · Answer

it is a l'hopital question

tkhunny · Answer

Have you considered a logarithm?

jennychan12 · Answer

yes, i took ln of both sides

anonymous · Answer

take the log, take the limit of the log, then exponentiate

tkhunny · Answer

??  There's only one side.

anonymous · Answer

or rewrite as $$x^{x^2}=e^{x^2\ln(x)}$$ then take the limit of that one
the work is all the same, finding 
$$\lim_{x\to 0}x^2\ln(x)$$ which you usually do by rewriting as 
$$\lim_{x\to 0}\frac{\ln(x)}{\frac{1}{x^2}}$$ to get it in the form $\frac{\infty}{\infty}$ so you can use l'hopital

jennychan12 · Answer

ok, thanks :)

agent0smith · Answer

You can somewhat evaluate it analytically, since the x^2 will approach 0 faster than the x, so it'll approach 1. Wolfram uses a similar method to @satellite73 http://www4a.wolframalpha.com/Calculate/MSP/MSP4614217a9acbb15hadbg0000230hc10ec765i6f0?MSPStoreType=image/png&s=31&w=492&h=938