f(θ) = {(1 - tan θ)/(1 - √2 sin θ), for x ≠ π/4, k/2, for x = π/4 at θ = π/4

Question

i. \(f(θ) = \begin{cases} \frac{1-tan\,\theta}{1-\sqrt{2}\,sin\,\theta}, & \quad \text{for } x\neq \frac{\pi}{4}\\ \frac{k}{2}, & \quad \text{for } x=\frac{\pi}{4} \end{cases}\) at θ = \(\frac{\pi}{4}\)

f(θ) = {(1 - tan θ)/(1 - √2 sin θ), for x ≠ π/4, k/2, for x = π/4 at θ = π/4

ii. \(f(θ) = \begin{cases} \left[tan\left(\frac{\pi}{4}+x \right)\right]^{\frac{1}{x}}, & \quad \text{for } x\neq 0\\ k, & \quad \text{for } x=0 \end{cases}\) at x = 0

Answer 1

i. \(f\left(\frac{\pi}{4}\right) = \frac{k}{2}\) .......(given)

Since f(x) is continuous at θ = \(\frac{\pi}{4}\)

ii. f(0) = k ….(given)

Since f(x) is continuous at x = 0