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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\text{k}\text{x}^2,&\text{x}\geq1\\4,&\text{x}<1\end{cases}\text{at x} =1$
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Answer

Given,
$\text{f(x)}=\begin{cases}\text{k}\text{x}^2,&\text{x}\geq1\\4,&\text{x}<1\end{cases}$
We have,
$(\text{LHL at x}= 1)=\lim_\limits{\text{x}\rightarrow1^-}\text{f(x)}=\lim_\limits{\text{h}\rightarrow0}\text{f}(1-\text{h})$
$\lim_\limits{\text{h}\rightarrow0}4=4$
$(\text{RHL at x}= 1)=\lim_\limits{\text{x}\rightarrow1^+}\text{f(x)}=\lim_\limits{\text{h}\rightarrow0}\text{f}(1+\text{h})$
$\lim_\limits{\text{h}\rightarrow0}\text{k}(1+\text{h})^2=\text{k}$
If f(x) is continuous at x = 1, then
$\lim_\limits{\text{x}\rightarrow1^-}\text{f(x)}=\lim_\limits{\text{x}\rightarrow1^+}\text{f(x)}$
$\Rightarrow\text{k}=4$

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