Evaluate the following limit. If you use LHospital

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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⋅21−x)  =3⋅limx→∞(21−x)+2⋅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⋅lim(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, then ∑n=1∞|an| converges.  If |an+1an|&amp;gt;1 eventually then ∑n=1∞an diverges. limx→∞(x22x)=0  =2⋅0  =0  limx→∞(3⋅21−x+x2⋅21−x)=3⋅limx→∞(21−x)+2⋅limx→∞(x212x)  =0+0  =0  Result: limx→∞(3⋅21−x+x2⋅21−x)=0

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})

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