Indeterminate forms 0^0 and 1^\infty. Evaluate the following limits.

xcl3411 2021-11-20 Answered
Indeterminate forms 00 and 1. Evaluate the following limits.
a) limx0+xx
b) limx(1+1x)x
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Wasither1957
Answered 2021-11-21 Author has 17 answers
Given to find:
a)limx0+(x)x
b) limx(1+1x)x
Part a:
Since u=eln(u), then:
limx0+xx=limx0+eln(xx)
limx0+eln(xx)=limx0+exln(x)
Move the limit under the exponential:
limx0+exln(x)=elimx0+exln(x)
elimx0+xln(x)=elimx0+ln(x)1x
Since we have an indeterminate form of type , we can apply the LHospital
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juniorekze
Answered 2021-11-22 Author has 18 answers
Part b:
Since u=eln(u), then:
limx(1+1x)x=limxeln((1+1x)x)
limxeln((1+1x)x)=limxexln(1+1x)
Move the limit under the exponential:
limxexln(1+1x)=elimxxln(1+1x)
elimxxln(1+1x)=elimxln(1+1x)1x
Since we have an indeterminate form type 00, we can apply the LHopitals rule:
elimxln(1+1x)1x=elimxddxln(1+1x)ddx1x
elimxddxln(1+1x)ddx1x=elimx11+1x
elimx11+1x=elimxxx+1
elimxxx+1=elimx
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