Question

If f(x) = `(sqrt(2) cos x - 1)/(cot x - 1), x ≠ pi/4` find the value of `"f"(pi/4)`  so that f (x) becomes continuous at x = `pi/4`

Answer 1

Given, f(x) = `(sqrt(2) cos x - 1)/(cot x - 1), x ≠ pi/4`

Therefore, `lim_(x -> pi/4) "f"(x) = lim_(x -> pi/4) (sqrt(2) cos x - 1)/(cot x - 1)`

= `lim_(x -> pi/4) ((sqrt(2) cos x - 1) sin x)/(cos x - sin x)`

= `lim_(x -> pi/4) ((sqrt(2) cos x - 1))/((sqrt(2) cos x + 1)) * ((sqrt(2) cos x + 10))/((cosx - sin x)) * ((cosx + sin x))/((cos x + sin x)) * sin x`

= `lim_(x -> pi/4) (2cos^2 x - 1)/(cos^2 x - sin^2x) * (cosx + sinx)/(sqrt(2) cos x + 1) * (sin x)`

= `lim_(x -> pi/4) (cos 2x)/(cos 2x) * ((cosx + sinx)/(sqrt(2) cos x + 1)) * (sin x)`

= `lim_(x -> pi/4) ((cosx + sin x))/(sqrt(2) cos x + 1) sinx`

= `(1/sqrt(2) (1/sqrt(2) + 1/sqrt(2)))/(sqrt(2) * 1/sqrt(2) + 1)`

= `1/2`

Thus, `lim_(x -> pi/2) "f"(x) = 1/2`

If we define `"f"(pi/4) = 1/2`, then f(x) will become continuous at x = `pi/4`.

Hence for f to be continuous at x = `pi/4`, `"f"(pi/4) = 1/2`.