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CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
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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