Extend the definition of the following by continuity fx-1-cos7x-5x-2 at the point x-

Given, f(x)=1 cos 7(x π)5(x π)^2 lim_x⇒π1 cos 7(x π)5(x π)^2=lim_x⇒π1 cos (7x 7π)5(x π)^2 = x →πlim2 sin^2 72 (x π)5.(x π)^2 =2×49× (x π ...

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Theorems for Continuity

Extend the de...

Question

Extend the definition of the following by continuity $f (x) = \frac{1 - cos 7 (x - π)}{5 {(x - π)}^{2}}$ at the point $x = π$ .

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f (x) = \frac{1 - \cos 7 (x - π)}{5 (x - π)^{2}}

f (x) = \frac{1 - cos (7 (x - π))}{5 (x - π)^{2}},

x \neq π

x = π,

f (π)

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} - 2 sin x, & - π \leq x \leq \frac{- π}{2} a cos (x) + b, & \frac{- π}{2} < x < 0 - 2 + cos x, & 0 \leq x \leq \frac{π}{2} c sin x, & \frac{π}{2} < x \leq π \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

[- π, π],

f (x)

α

β

f (x)

f (x) = - 2 sin x,

- π \leq x \leq - \frac{π}{2}

= α sin x + β,

- \frac{π}{2} < x < \frac{π}{2}

= cos x,

\frac{π}{2} \leq x \leq π

[- π, π]

f (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{1 - sin x}{(π - 2 x)^{2}} \cdot \frac{log sin x}{log (1 + π^{2} - 4 π x + 4 x^{2})}, & x \neq \frac{π}{2} k, & x = \frac{π}{2} \end{matrix}

x = \frac{π}{2}

k

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