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 \(\frac{-2}{\pi}\).