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Question
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}kx + 5, & \text{ if } x \leq 2 \\ x - 1, & \text{ if } x > 2\end{cases}\]
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Solution
Given:
\[ \Rightarrow \lim_{h \to 0} \left( k\left( 2 - h \right) + 5 \right) = \lim_{h \to 0} \left( 2 + h - 1 \right)\]
\[ \Rightarrow 2k + 5 = 1\]
\[ \Rightarrow 2k = - 4\]
\[ \Rightarrow k = - 2\]
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