Show that the following functions have a continuous extension to the point where f(x) is not defined. f(x) = 1-cos 2x/ sin x, for x ≠ 0.

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) = 1-cos 2x/ sin x, for x ≠ 0.

Answer 1

f(x) = 1-cos 2x/ sin x, 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) = 1-cos 2x/ sin x, for x ≠ 0.

= 0 for x = 0 

∴ f(x) is continuous at x = 0.