Evaluate the following limits. If you use lHospitals Rule, be sure to indicate when yyou are using it, and why it applies. a) limx→∞(3⋅21−x+x2⋅21−x) b) limx→0+(1+5x)2x

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Evaluate the following limits. If you use lHospitals Rule, be sure to indicate when yyou are using it, and why it applies.  a) limx→∞(3⋅21−x+x2⋅21−x)  b) limx→0+(1+5x)2x

Walker Funk · Accepted Answer

Evaluate limits:  limx→∞(3⋅21−x+x2⋅21−x)  limx→a[f(x)±g(x)]=limx→af(x)±limx→ag(x)  limx→∞(3⋅21−x+x2⋅21−x)=limx→∞(3⋅21−x)+limx→∞(x2⋅2−x)  =3⋅limx→∞(21−x)+2⋅limx→∞(x212x)  Take 3⋅limx→∞(21−x)  Apply exponent rule ax=eln⁡(ax)=exln⁡(a)  =3⋅limx→∞(e(1−x)ln⁡(2)))  =3⋅limx→∞(21−x)  =3⋅limx→−∞(exln⁡(2))  Apply the Limit Chain Rule:  g(x)=xln⁡(2),f(u)=eu  =−∞⋅ln⁡(2)  =−∞  3⋅limx→−∞(exln⁡(2))=3⋅e−∞  =3⋅0  =0  Take 2⋅limx→∞(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&amp;lt;q&amp;lt;1, the

Lauren Fuller · Accepted Answer

b) Evaluate limits:  limx→0+(1+5x)2x  Apply exponent rule:  ax=eln⁡(ax)=ex⋅ln⁡(a)  (1+5x)2x=e2xln⁡(1+5x)  Apply the limit Chain Rule:  g(x)=2xln⁡(1+5x),f(u)=eu  limx→0+g(x)=limx→0+2xln⁡(1+5x)  =2⋅limx→0+(ln⁡(1+5x)x)  limx→0+(ln⁡(1+5x)x)=00  LHopital

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

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

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