Find the value of k so that the function f is continuous at the i… - Vidyadip

Question

Find the value of $k$ so that the function $f$ is continuous at the indicated point:
$f(x)=\left\{\begin{array}{l}x+1 \text { if } x \leq \pi \\ \cos x \text { if } x>\pi\end{array}\right.$ at $x=\pi$

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Answer

Left hand limit $= \mathop {\lim }\limits_{x \to {\pi ^ - }} f(x) = \mathop {\lim }\limits_{x \to {\pi ^ - }} \left( {Kx + 1} \right)$
$= \mathop {\lim }\limits_{h \to 0} \left[ {K\left( {\pi - h} \right) + 1} \right]$
$= K\pi + 1$
Right hand limit $= \mathop {\lim }\limits_{x \to {\pi ^ + }} f(x) = \mathop {\lim }\limits_{x \to {\pi ^ + }} \cos x$
$= \mathop {\lim }\limits_{h \to 0} \cos \left( {\pi + h} \right) = \mathop {\lim }\limits_{h \to 0} - \cos \,h$
$= -\cos 0 = - 1$
Therefore,
$K\pi + 1 = - 1$
$\Rightarrow K = \frac{{ - 2}}{\pi }$

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