What is #\lim _ { x \rightarrow 0} x \ln x#?

1 Answer
Teddy
Jul 2, 2018

Since the domain of #lnx# is #(0,oo)#, it only makes sense to take #lim_(xrarr0^+)xlnx# instead of #lim_(xrarr0)xlnx# since it doesn't exist. #lim_(xrarr0^+)xlnx=0#.

Explanation:

#L=lim_(xrarr0^+)xlnx#
#L=lim_(xrarr0^+)lnx/(1/x)#
Since this is an indeterminate form, apply L'Hopital's rule to get
#L=lim_(xrarr0^+)(1/x)/(-1/x^2)#
#L=lim_(xrarr0^+)-x#
#:.L=0#

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