The value of k, if f(x) = (√2 cos x − 1)/(cot x − 1) for x ≠ π/4 and f(π/4) = k, so that f is continuous at x = π/4, is:

Explanation

Continuity at x = π/4 requires k = lim x→π/4 [(√2 cos x − 1)/(cot x − 1)]. Both numerator and denominator approach 0. Applying L'Hôpital's rule gives k = [−√2 sin(π/4)]/[−csc²(π/4)] = (−1)/(−2) = 1/2.