Question

In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}kx + 5, & \text{ if  }  x \leq 2 \\ x - 1, & \text{ if }  x > 2\end{cases}\]

Answer 1

Given: 

\[f\left( x \right) = \begin{cases}kx + 5, & \text{ if  }  x \leq 2 \\ x - 1, & \text{ if }  x > 2\end{cases}\]

If  \[f\left( x \right)\] is continuous at x = 2, then 

\[\lim_{x \to 2^-} f\left( x \right) = \lim_{x \to 2^+} f\left( x \right)\]

\[\Rightarrow \lim_{h \to 0} f\left( 2 - h \right) = \lim_{h \to 0} f\left( 2 + h \right)\]
\[ \Rightarrow \lim_{h \to 0} \left( k\left( 2 - h \right) + 5 \right) = \lim_{h \to 0} \left( 2 + h - 1 \right)\]
\[ \Rightarrow 2k + 5 = 1\]
\[ \Rightarrow 2k = - 4\]
\[ \Rightarrow k = - 2\]