Let fx- 1-sin 3 x 3cos 2 x - if x- 2- If fx is continuous at x-2- find a and b-

x →π2^ lim #160; f(x) = f(π2) = #160; x →π2^+lim f(x) x →π2^ lim1 sin^3x3 cos^2 x = a ⇒x →π2^ lim(1 sin x) (1 + sin^2 x + sin x)3 (1 ...

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Theorems for Continuity

Let fx= 1-...

Question

Let $f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{1 - {sin}^{3} x}{3 {cos}^{2} x}, i f x < \frac{π}{2} a, i f x = \frac{π}{2} \frac{b (1 - sin x)}{{(π - 2 x)}^{2}}, i f x > \frac{π}{2} \end{matrix}$ . If $f (x)$ is continuous at $x = \frac{π}{2}$ , find $a$ and $b$ .

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

f (x)

x = \frac{π}{2}

x = \frac{π}{2}

a

b

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

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

p

q

f (x) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{1 - {sin}^{3} x}{3 {cos}^{2} x}, i f x < \frac{π}{2} p, i f x = \frac{π}{2} i s c o n t i n o u s a t x = \frac{π}{2} \frac{q (1 - sin x)}{{(π - 2 x)}^{2}}, i f x > \frac{π}{2} \end{matrix}

p

q

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

\frac{π}{2}

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