Evaluate the following limits. lim_{xrightarrowinfty}(3cdot2^{1-x}+x^2cdot2^{1-x})

amanf 2021-02-09 Answered
Evaluate the following limits. limx(321x+x221x)
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Expert Answer

Derrick
Answered 2021-02-10 Author has 94 answers

Evaluate limit:
limx(321x+x221x)
limxa[f(x)+g(x)]=limxaf(x)+limxag(x)
limx(321x+x221x)=limx(321x)+limx(x221x)
=3limx(21x)+2limx(x22x)
=3limx(21x)+2limx(x212x)
Take 3limx(21x)
Apply exponent rule ax=elnax=exlna
=3limx(e(1x)ln2)
=3limx21x
=3limxexln2
Apple the Limit Chain Rule:
g(x)=xln(2),f(u)=eu
=ln2
=
3limxexln2=3e
=30
=0
Take 2limxx212x
if sum an converges,then limn(an)=0
Apply ratio test and check weather series is convergent of divergent
if |an+1an|q eventually for some 0, then n=1x|an| converges,
if |an+1an|>1 eventually then n=1xan diverges
limx(x22x)=0
=20
=0
limx(321x+x221x)=3limx(21x)+2limx(x212x)
=0+0
=0
Result:
3limx(21x)+2limx(x212x)=0

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