Question

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; 

\[f\left( x \right) = \begin{cases}kx + 1, \text{ if }  & x \leq \pi \\ \cos x, \text{ if }  & x > \pi\end{cases}\] at x = π

Answer 1

\[f\left( x \right) = \begin{cases}kx + 1, \text{ if }  & x \leq \pi \\ \cos x, \text{ if }  & x > \pi\end{cases}\] 

We have
(LHL at x =  \[\pi\] = \[\lim_{x \to \pi^-} f\left( x \right) = \lim_{h \to 0} f\left( \pi - h \right) = \lim_{h \to 0} k\left( \pi - h \right) + 1 = k\pi + 1\]

\[\pi\]= \[\lim_{x \to \pi^+} f\left( x \right) = \lim_{h \to 0} f\left( \pi + h \right) = \lim_{h \to 0} \cos\left( \pi + h \right) = cos\pi = - 1\]

If f(x) is continuous at x = 

\[\pi\] , then

\[\lim_{x \to \pi^-} f\left( x \right) = \lim_{x \to \pi^+} f\left( x \right)\]

\[\Rightarrow k\pi + 1 = - 1\]
\[ \Rightarrow k = \frac{- 2}{\pi}\]