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Question

Let f(x)=limn(cosxn)n, g(x)=limn(1x+x.ne)n
limx0(f(x))(g(x)) is equal to

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Solution

f(x)=limncos xnn

g(x)=limn(1x+x.ne)n

Taking log on both sides
lnf(x)=limnsinxncosxn.12xn.xn21n2

lnf(x)=limnsinxn.xcosxn.2xn

lnf(x)=12.limn(11)x.sinxnxn

lnf(x)=12limnsinxnxn.x

lnf(x)=x2......⎜ ⎜ ⎜ ⎜nxn0limxn0sinxnxn=1⎟ ⎟ ⎟ ⎟

f(x)=ex/21

Now g(x)=limn(1x+xne)n

Taking ln on both sides,
lng(x)=limnnlog(1x+x.e1/n)

lng(x)=limn1(1x+x.e1/n).(x.e1/n).1n21n2 (Applying L-Hospital rule)
lng(x)=(11).x.(1)1

lng(x)=x

g(x)=ex

limx0(f(x))(g(x))=limx0(ex/2)(ex)=x2x=12

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