Saturday, March 11, 2017

Single Variable Calculus, Chapter 7, 7.8, Section 7.8, Problem 60

Determine the limx(ex+x)1x. 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=(ex+x)1x, then
lny=(1x)ln(ex+x)

So,
limxlny=limxln(ex+x)x
By applying L'Hospital's Rule...
limxln(ex+x)x=limxex+1ex+x1=limxex+1ex+x

If we evaluate the limit, we will still get indeterminate form, so we next apply L'Hospital's Rule once more. Thus,
limxex+1ex+x=limxexex+1
Again, by applying L'Hospital's Rule...
limxexex+1=limxexex=1

Hence,
limxlny=limxln(ex+x)x=1

Therefore, we have
limx(ex+x)1x=limxelny=e1=e

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