For what value of k is the following function continuous at x = 2… - Vidyadip

Question

For what value of $ k$ is the following function continuous at $x = 2$

$f(x) = \begin{matrix} 2x + 1 & ; & x< 2 \\ k & ; & x = 2 \\ 3x - 1 & ; & x> 2 \end{matrix} $

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Answer

For continuity of the function at $x = 2$

$\lim\limits_{h \rightarrow 0} f (2 -h) = f(2) = \lim\limits_{h \rightarrow 0} f(2 + h)$

Now, $f (2 - h) = 2 (2 - h) + 1 = 5 - 2h$

$\therefore \lim\limits_{h \rightarrow 0} f(2- h) = 5$

Also, $f(2 + h) = 3(2 + h) -1 = 5 + 3h$

$ \lim\limits_{h \rightarrow 0} f(2 + h) = 5$

So, for continuity $f(2) = 5.$

$\therefore k = 5.$

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