Question

Find the values of k so that the function f is continuous at the indicated point.
f(x)=\left\\{\begin{matrix} kx+1, &if\, x\leq\pi \\\ cos\,x,&if\,x>\pi \end{matrix}\right.\,at\,x=\pi

Answer 1

The given function is

f(x)=\left\\{\begin{matrix} kx+1, &if\, x\leq\pi \\\ cos\,x,&if\,x>\pi \end{matrix}\right.

The given function f is continuous at x=p, if f is defined at x=p and if the value of the f at x=p equals the limit of f at x=p.
It is evident that f is defined at x=p and f(π)=kπ+1
$\lim_{x\rightarrow\pi^-}$ f(x)=$\lim_{x\rightarrow\pi^+}$f(x)=f($\pi$)
\Rightarrow$$\lim_{x\rightarrow\pi^-}(kx+1)=$\lim_{x\rightarrow\pi^+}$cosx=k$\pi$+1
$\Rightarrow$k$\pi$+1=cos$\pi$=k$\pi$+1
$\Rightarrow$k$\pi$+1=-1=k$\pi$+1
k=$\frac{-2}{\pi}$
Therefore, the required value of k is $\frac{-2}{\pi}$.