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Question

Find the value of k for which f(x)=1cos4x8x2,when x0k , when x=0 is continuous at x=0.

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Solution

Given,
f(x)=1cos4x8x2,when x0k,when x=0

f(0)=k

If f(x) is continuous at x=0, then

limx0f(x)=f(0)

limx01cos4x8x2=k

limx02sin22x8x2=k

22limx0sin22x4x2=k

limx0sin22x(2x)2=k

k=limx0(sin2x2x)2

k=1

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