I needed to solve
limx→0x∗lnx
and I wasn't sure how I would do it so I looked up the answer.
They used L'Hoptial to solve this and I don't understand why this works.
limx→0x∗lnx=limx→0lnx1/x but I can't use L'Hopital here because this is
undefined0, so I looked up if ln 0 is really undefined and it turns out that the limit of ln0 is −infinity
My textbook says I can only use L'Hopital with infinf or 00, so why am I allowed to use L'Hopital in this case?
Answer
You can use L'Hopital if you rewrite your expression lnx1/x
As x→0+, we have the indeterminate form of ∞∞, so you are now licensed to take the derivative of the numerator and of the denominator and evaluate the limit.
limx→0+1x−1x2=−limx→0+x=0
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