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Question
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}2 , & \text{ if } x \leq 3 \\ ax + b, & \text{ if } 3 < x < 5 \\ 9 , & \text{ if } x \geq 5\end{cases}\]
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Solution
Given:
\[\lim_{x \to 3^-} f\left( x \right) = \lim_{x \to 3^+} f\left( x \right) \text{ and } \lim_{x \to 5^-} f\left( x \right) = \lim_{x \to 5^+} f\left( x \right)\]
\[\Rightarrow \lim_{h \to 0} f\left( 3 - h \right) = \lim_{h \to 0} f\left( 3 + h \right) \text{ and } \lim_{h \to 0} f\left( 5 - h \right) = \lim_{h \to 0} f\left( 5 + h \right) \]
\[ \Rightarrow \lim_{h \to 0} \left( 2 \right) = \lim_{h \to 0} \left( a\left( 3 + h \right) + b \right) \text{ and } \lim_{h \to 0} \left( a\left( 5 - h \right) + b \right) = \lim_{h \to 0} \left( 9 \right)\]
\[ \Rightarrow 2 = 3a + b \text{ and } 5a + b = 9\]
\[ \Rightarrow 2 = 3a + b \text{ and } 5a + b = 9\]
\[ \Rightarrow a = \frac{7}{2} \text{ and } b = \frac{- 17}{2}\]
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