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Continuity — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity2 Marks
Question
In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\text{k}(\text{x}^2+2),&\text{if x}\leq0\\3\text{x}+1,&\text{if x}>0\end{cases}$
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Answer

$\text{f(x)}=\begin{cases}\text{k}(\text{x}^2+2),&\text{if x}\leq0\\3\text{x}+1,&\text{if x}>0\end{cases}$
We know that a function will be continuous x = 0. if
$\text{LHL}=\text{RHL}=\text{f}(0)\ ....(\text{i})$
$\text{f}(0)=\text{k}(0+2)=2\text{k}$
$\text{LHL}=\lim_\limits{\text{x}\rightarrow0^+}\text{f(x)}=\lim_\limits{\text{h}\rightarrow0}(0+\text{h})$
$=\lim_\limits{\text{h}\rightarrow0}3(\text{h})+1=1$
Thus, using (i) we get,
$2\text{k}=1$
$\text{k}=\frac{1}{2}$

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