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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`
Solution
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`.
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