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antennense 2022-07-13 Answered
Evaluate lim x e x 2 x x + ln ( x ) / x e t 2 d t
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Answers (1)

soosenhc
Answered 2022-07-14 Author has 16 answers
First, we'll use a limit law, saying the limit of a product is a product of the limits.
lim x e x 2 lim x x x + ln ( x ) x e t 2 d t
Let's evaluate the first limit.
lim x e x 2 = lim x 1 e x 2
As x approaches , e x 2 also approaches , therefore 1 e x 2 approaches 0, therefore one of the limits calculated is 0.
However, we're not done. If the limit of the integral is , we would get an indeterminate form. By monotonicity,
| x x + ln ( x ) x e t 2 d t | ( x + ln ( x ) x x ) e x 2 = ln ( x ) x e x 2
, which if we evaluate as x approaches , we get 0 0 = 0, so we didn't have to use L'Hôpital's rule.
Hope this helps!

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