Find the limit of \(\displaystyle{\left({e}^{{{2}{x}}}+{1}\right)}^{{\frac{{1}}{{x}}}}\)

Aileen Rogers 2022-04-12 Answered
Find the limit of (e2x+1)1x
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Answers (1)

shanna87mn
Answered 2022-04-13 Author has 19 answers
We have e2x<e2x+1<e2x+e2x for x>0. Hence, we get
(e2x)1x<(e2x+1)1x<(e2x+e2x)1x
i.e.
e2<(e2x+1)1x<(2e2x)1x=21xe2
Now take the limit and note that limx21x=1 to get that
e2limx(e2x+1)1xe2
Hence, we get that
limx(e2x+1)1x=e2
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