Evaluate the following limit. If you use L'Hospital Rule, be sure to indicate when you are using it, and why it applies. lim_{xrightarrowinfty}(3cdot2^{1-x}+x^2cdot2^{1-x})

illusiia 2020-12-22 Answered
Evaluate the following limit. If you use LHospital
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Expert Answer

okomgcae
Answered 2020-12-23 Author has 93 answers
Evaluate limits:
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=eln(ax)=exln(a)
=3limx(e(1x)ln(2))
=3limx(21x)
=3limx(exln(2))
Apply the Limit Chain Rule:
g(x)=xln(2),f(u)=eu
=ln(2)
=
=3lim(exln(2))=3e
=30
=0
Take 2limx(x212x
if an converges, then limn(an)=0
Apply ratio test and check weather series is convergent or divergent.
If |an+1an|q eventually for some 0<q<1, then n=1|an| converges.
If |an+1an|>1 eventually then n=1an
diverges. limx(x22x)=0
=20
=0
limx(321x+x221x)=3limx(21x)+2limx(x212x)
=0+0
=0
Result: limx(321x+x221x)=0
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