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Continuity — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity3 Marks
Question
Determine the value of the constant k so that the function $\text{f(x)}=\begin{cases}\text{kx}^2,&\text{if }\text{ x}\leq2\\3,&\text{if }\text{ x}>2\end{cases}$ is continuous at x = 2.
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Answer

Given, $\text{f(x)}=\begin{cases}\text{kx}^2,&\text{if }\text{ x}\leq2\\3,&\text{if }\text{ x}>2\end{cases}$

If f(x) is continuous at x = 2, then

$\lim\limits_{\text{x}\rightarrow2^-}\text{f(x)}=\lim\limits_{\text{x}\rightarrow2^+}=\text{f}(2)\ ...(\text{i})$

Now,

$\lim\limits_{\text{x}\rightarrow2^-}\text{f(x)}=\lim\limits_{\text{h}\rightarrow0}\text{f}(2-\text{h})\\=\lim\limits_{\text{h}\rightarrow0}\text{k}(2-\text{h})^2=4\text{k}$

And, f(2) = 3

From (i) we have,

$4\text{k}=3$

$\Rightarrow\text{k}=\frac{3}{4}$

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