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Lt_x→π21 sin x(π 2x)^2 (L Hospital) Lt_x→π2 cos x2(π 2x) 2 (L Hospital) Lt_x→π2 sin x4x 2⇒18 f(π2)=18 ...

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Continuity of a Function

If fx=1-sin...

Question

If $f (x) = \frac{1 - sin x}{(π - 2 x)^{2}}$ , for $x \neq \frac{π}{2}$ is continous at $x = \frac{π}{2}$ , find $f (\frac{π}{2}) .$

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f (x) = \frac{\sqrt{2} - \sqrt{1 + s i n x}}{c o s x}

x \neq π / 2

x = π / 2

f (π / 2)

f (x) = \frac{1 - sin x}{sin 2 x}, x \neq \frac{π}{2}

f (x)

x = \frac{π}{2}

f (\frac{π}{2})

p

q

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{1 - sin^{3} x}{{cos}^{2} x}, i f x < π / 2 p, i f x = π / 2 \frac{q (1 - sin x)}{(π - 2 x)^{2}}, i f x > π / 2 \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

x = π / 2

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{(1 - {sin}^{3} x)}{3 {cos}^{2} x}, & x < \frac{π}{2} a, & x = \frac{π}{2} \frac{b (1 - sin x)}{(π - 2 x)^{2}}, & x > \frac{π}{2} \end{matrix}

x = \frac{π}{2}

{(\frac{b}{a})}^{5 / 3}

x = π

f (x) = \frac{\sqrt{2 + c o s x - 1}}{(π - x^{2})}

x \neq π

f (π)

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