Find the values of k so that the function f is continuous at the indicated point.


Find the values of so that the function f is continuous at the indicated point.

$f(x)=\left\{\begin{array}{l}k x+1, \text { if } x \leq \pi \\ \cos x, \text { if } x>\pi\end{array} \quad\right.$ at $x=\pi$



The given function is $f(x)= \begin{cases}k x+1, & \text { if } x \leq \pi \\ \cos x, & \text { if } x>\pi\end{cases}$

The given function f is continuous at x = p, if f is defined at x = p and if the value of f at x = p equals the limit of f at x = p

It is evident that $f$ is defined at $x=\mathrm{p}$ and $f(\pi)=k \pi+1$

$\lim _{x \rightarrow \pi^{-}} f(x)=\lim _{x \rightarrow \pi^{+}} f(x)=f(\pi)$

$\Rightarrow \lim _{x \rightarrow \pi^{-}}(k x+1)=\lim _{x \rightarrow x^{+}} \cos x=k \pi+1$

$\Rightarrow k \pi+1=\cos \pi=k \pi+1$

$\Rightarrow k \pi+1=-1=k \pi+1$

$\Rightarrow k=-\frac{2}{\pi}$

Therefore, the required value of $k$ is $-\frac{2}{\pi}$.

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