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Question

If limx0f(x)x=10 and limx0f(sin x)g(x) cos2 x=5 with g(0)=0, then the value of limx0g(1cos x)x2is

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Solution

limx0f(x)x=10
So, f(0) must be 0.

limx0f(x)x (Indeterminate form)
Using L' Hospitals' rule, we get
limx0f(x)1=10
f(0)=10 ...(1)

Now, limx0f(sin x)g(x) cos2 xIndeterminate form
Using L' Hospitals' rule, we get
limx0f(sin x)×cos xg(x)cos2 x+g(x)×2 cos x×(sin x)=5

f(0)g(0)=5g(0)=105=2 ...(2)
limx0g(1cos x)x2 Indeterminate form

=limx0g(1cos x)×sin x2x=g(0)2=22=1

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