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Substitution Method to Remove Indeterminate Form

Let f x = [...

Question

Let $f (x) = \begin{matrix} \frac{k cos x}{π - 2 x} & w h e n x \neq \frac{π}{2} 3 & x = \frac{π}{2} \end{matrix}$ and if $lim x \to \frac{π}{2} f (x) = f (\frac{π}{2})$ .
find the value of`k.

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Solution

$f (x) = \frac{k cos x}{π - 2 x}$ , when $x \neq \frac{π}{2}$
$= 3$ when $x = \frac{π}{2}$

Now $l i m x \to \frac{π}{2} f (x) = 3$

$\Rightarrow l i m x \to \frac{π}{2} f (x) = \frac{K cos x}{π - 2 x} = 3$

Now $l i m x \to \frac{π}{2} f (x) | \frac{0}{0}$ from

$l i m x \to \frac{π}{2} f (x) |$ Applying L' Hospitalize Rule

$= \frac{- K}{- 2} = \frac{K}{2}$

$∴ \frac{K}{2} = 3$

$\Rightarrow K = 6$

$∴ K = 6$

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f (x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} \frac{k cos x}{π - 2 x}, if x \neq \frac{π}{2} 3, if x = \frac{π}{2} \end{matrix}

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