How do you find the limit of x^(x^2) as x approaches 0?

#x^(x^2)#

1 Answer
Ultrilliam
Apr 21, 2018

See below.

Explanation:

#lim_(x to 0) x^(x^2)#

#= lim_(x to 0) exp ( ln( x^(x^2)) )#

#= lim_(x to 0) exp (x^2 ln( x) )#

#= lim_(x to 0) exp (ln( x)/(1/x^(2)) )#

The exponential function is continuous through the limit, and #(ln( x)/(1/x^(2)) ) # is in indeterminate #oo/oo# form, so we can apply L'Hôpital's Rule, within the exponent:

#= lim_(x to 0) exp ((1/x)/(- (2)/x^(3)) )#

#= lim_(x to 0) e^ (- x^2/2 ) = 1#

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