Solve this question \lim_{x\to \infty} (x+3)^{1 + 1/x} - x^{1 +

Ali Marshall 2022-04-28 Answered
Solve this question
limx(x+3)1+1xx1+1x+3
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Answers (1)

Tyler Velasquez
Answered 2022-04-29 Author has 19 answers
Write the function of interest as f(x)=×1xg(x) with
g(x)=(1+3x)1+1xx1x+31x=ea(x)eb(x)
with
a(x)=(1+1x)log(1+3x)=3x+o(1x)
and
b(x)=3log(x)x(x+3)=o(1x)
This yields ea(x)=1+3x+o(1x) and eb(x)=1+o(1x) hence g(x)=3x+o(1x)
Likewise x1x=elog(x)x=1+o(1) hence
f(x)=×1xg(x)=x(1+o(1))(3x+o(1x))=3+o(1)
That is, f does converge and its limit is 3
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