Show that the following functions have a continuous extension to the point where f(x)= 3sin^2x+2 cos x (1-cos 2x)/ 2(1-cos^2x)

Question

Show that the following functions have a continuous extension to the point where f(x) is not defined. Also, find the extension.

f(x) = \(\frac {3\, sin^2x+2\,cos\,x(1-cos\,2x)}{2(1-cos^2x)}\), for x ≠ 0.

f(x) = 3sin²x+2 cos x (1-cos 2x)/ 2(1-cos²x)

Answer 1

f(x) = \(\frac {3\, sin^2x+2\,cos\,x(1-cos\,2x)}{2(1-cos^2x)}\), for x ≠ 0.

Here, f(0) is not defined. 

Consider,

But f(0) is not defined. 

∴ f(x) has a removable discontinuity at x = 0. 

∴ The extension of the original function is

f(x) = \(\frac {3\, sin^2x+2\,cos\,x(1-cos\,2x)}{2(1-cos^2x)}\), x ≠ 0.

= 7/2, x = 0

∴ f(x) is continuous at x = 0.