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Continuity — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsContinuity5 Marks
Question
Find the values of a so that the function
$\text{f}\text{(x)}=\begin{cases}\text{ax}+5, &\text{if}\text{ x}\leq2\\\text{x}-1, &\text{if}\text{ x}>2\end{cases}$ is continuous at x = 2.
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Answer

Given,
$\text{f}\text{(x)}=\begin{cases}\text{ax}+5, &\text{if}\text{ x}\leq2\\\text{x}-1, &\text{if}\text{ x}>2\end{cases}$
We observe
$\text{(LHL at x}=2)=\lim\limits_{\text{x} \rightarrow 2^-}\text{f}\text{(x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}(2-\text{h)}$
$=\lim\limits_{\text{h} \rightarrow 0}\text{a}(2-\text{h)}+5=2\text{a}+5$
$\text{(RHL at x}=2)=\lim\limits_{\text{x} \rightarrow 2^+}\text{f}\text{(x)}=\lim\limits_{\text{h} \rightarrow 0}\text{f}(2+\text{h)}$
$=\lim\limits_{\text{h} \rightarrow 0}(2+\text{h}-1)=1$
And, $\text{f}(2)=\text{a}(2)+5=2\text{a}+5$
Since f(x) is continuous at x = 2, we have
$=\lim\limits_{\text{x} \rightarrow 2^-}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow 2^+}\text{f}\text{(x)}=\text{f}(2)$
$\Rightarrow2\text{a}+5=1$
$\Rightarrow2\text{a}=-4$
$\Rightarrow\text{a}=-2$

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