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Question

f(x) and g(x) are continuous functions such that limxa[3f(x)+g(x)]=6 and limxa[2f(x)g(x)]=4. Given that the function h(x).g(x) is continuous at x = a and h(a)=4, which of the following must be true?


A

limxah(x) ϵ R and limxa+h(x) ϵ R

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B

limxah(x)=0

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C

limxah(x)=

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D

limxah(x)=4

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Solution

The correct option is A

limxah(x) ϵ R and limxa+h(x) ϵ R


Given that
limxa[3f(x)+g(x)]=6.........(i)
limxa[2f(x)g(x)]=4...........(ii)
(i)+(ii)limxa[5f(x)]=10limxaf(x)=2limxag(x)=0=g(a) (g(x) is continuous)
Since h(x).g(x) is continuous,
limxah(x).limxag(x)=h(a).g(a)limxah(x)×0=4×0
This is true if limxah(x).


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