Tuesday, April 15, 2014

Single Variable Calculus, Chapter 7, 7.8, Section 7.8, Problem 58

Determine the limxxln21+lnx. Use L'Hospital's Rule where appropriate. Use some Elementary method if posible. If L'Hospitals Rule doesn't apply. Explain why.

If we let y=xln21+lnx, then
lny=(ln21+lnx)lnx

So,
limxlny=limx(ln2(lnx)1+lnx)

By applying L'Hospital's Rule...

limx(ln2(lnx)1+lnx)=limxln2(1x)0+(1x)=limxln2=ln2


Thus,

limxlny=limx(ln2(lnx)1+lnx)=ln2
Therefore, we have
limxxln21+lnx=limxelny=eln2=2

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