Class 12
MATHS
Continuity And Differentiability

# f(x) = {{:((1-coskx)/(x sinx), if x ne 0),(1/2, if x = 0):} at x = 0

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Updated On: 7-9-2020

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Very Important Questions

Text Solution

We have, f(x) = {{:((1-coskx)/(x sinx), if x ne 0),(1/2, if x = 0):} at x = 0  <br> At x=0, LHL = underset(hrarr0^(-))(lim)(1-coskx)/(xsinx)=underset(hrarr0)lim(1-cos(0-h))/((0-h)sin(0-h)) <br> <br> = underset(hrarr0)(lim)(1-cos(-kh))/(-hsin(-h)) <br> = underset(hrarr0)(lim)(1-coskh)/(hsinh) , [:' cos (-theta)= cos theta, sin (-theta)= - sin theta] <br> = underset(hrarr0)lim(1-1+2sin^(2)'(kh)/(2))/(hsinh), [:' cos theta = 1 - 2 sin^(2) ' (theta)/(2)] <br> = underset(hrarr0)(lim) (2sin^(2)'(kh)/(2))/(h sin h) <br> = underset(hrarr0)lim(2sin'(kh)/(2))/((kh)/(2)).(sin'(kh)/(2))/((kh)/(2)).(1)/((sinh)/(h)).(k^(2)h//4)/(h) <br> = (2k^(2))/(4) = (k^(2))/(2) , [:' underset(hrarr0)(lim)' (sinh)/(h) = 1] <br> Also, f(0) = 1/2 rArr (k^(2))/(2) = 1/2 rArr k +- 1
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