Solve this question \(\displaystyle\lim_{{{x}\to\infty}}{\left({x}+{3}\right)}^{{{1}+\frac{{1}}{{x}}}}-{x}^{{{1}+\frac{{1}}{{{x}+{3}}}}}\)

Deegan Chase

Deegan Chase

Answered question

2022-03-21

Solve this question
limx(x+3)1+1xx1+1x+3

Answer & Explanation

Alannah Farmer

Alannah Farmer

Beginner2022-03-22Added 11 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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