Evaluate the following limits. If you use l'Hospital's Rule, be sure to indicate

puntgewelb5 2021-11-21 Answered
Evaluate the following limits. If you use lHospitals Rule, be sure to indicate when yyou are using it, and why it applies.
a) limx(321x+x221x)
b) limx0+(1+5x)2x
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Expert Answer

Walker Funk
Answered 2021-11-22 Author has 13 answers
Evaluate limits:
limx(321x+x221x)
limxa[f(x)±g(x)]=limxaf(x)±limxag(x)
limx(321x+x221x)=limx(321x)+limx(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)
=
3limx(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, the
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Lauren Fuller
Answered 2021-11-23 Author has 14 answers
b) Evaluate limits:
limx0+(1+5x)2x
Apply exponent rule:
ax=eln(ax)=exln(a)
(1+5x)2x=e2xln(1+5x)
Apply the limit Chain Rule:
g(x)=2xln(1+5x),f(u)=eu
limx0+g(x)=limx0+2xln(1+5x)
=2limx0+(ln(1+5x)x)
limx0+(ln(1+5x)x)=00
LHopital
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