Examination the function `f(t)` given by `f(x)={(cosx)/(pi/2-x)`; for continuity at `x=pi/2`

Answer

Continuous

Explanation

$$ \begin{aligned} &f(x)=\frac{\cos x}{\frac{\pi}{2}-x} \quad \text { at } x=\pi / 2 \\ &\left.f\left(\pi / 2^{-}\right)=\frac{f(\pi / 2+)}{f(t)}=f / 2\right) \\ &\operatorname{lt}_{x \rightarrow \pi / 2} \frac{\cos x}{\frac{\pi}{2}-x} \quad x \rightarrow t+\pi / 2 \\ &\operatorname{lt}_{t \rightarrow 0} \frac{\cos (t+\pi / 2)}{-t} \Rightarrow t t-\frac{\sin t}{-t} \Rightarrow t \end{aligned} $$
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