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Question

The function is defined by f(x) = {kx+1,if xπcos x, if x>π at x = π.

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Solution

Here, f(x) = {kx+1,if xπcos x, if x>π


LHL = limxπf(x)=limxπ(kx+1)

Putting x=π-h as xπ when h0

limh0 k(πh)+1=limh0 kπkh+1=kπ+1

RHL = limxπ+f(x)=limxπ+cosx.

Putting x=π-h as xπ+ when h0

limh0 cos(πh)=limh0 cos h=1

Also, f(π) = (kπ)+1[(x)=kx+1]

Since, f(x) is continuous at x=π.

LHL=RHL=f(π)k(π)+1=k=2π


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