For what value of $k$ is the following function continuous at $x = 2$ ?
$f (x) = ⎧ ⎨ ⎩ \begin{matrix} \begin{matrix} 2 x + 1, k, \end{matrix} 3 x - 1, \end{matrix} \begin{matrix} if x < 2 if x = 2 if x > 2 \end{matrix}$

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Solution

R.H.L. $lim x \to 2 f (x) = lim x \to 2 (3 x - 1) = 5$
L.H.L, $lim x \to 2 f (x) = lim x \to 2 (2 x + 1) = 5$
$f (2) = k$
For f to be continuous, R.H.L. $=$ L.H.L $= f (2) \Rightarrow k = 5$ .

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