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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits3 Marks
Question
Evaluate $\lim\limits_{\text{x}\rightarrow2}{\text{f(x)}}$ (if it exist), where $\text{f(x)}=\begin{cases}\text{x}-[\text{x}],&\text{x}<2\\4 ,& \text{x} = 2\\3\text{x}-5, & \text{x} > 2\end{cases}.$
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Answer

$\lim\limits_{\text{x}\rightarrow2^-}{\text{f(x)}}=\lim\limits_{\text{x}\rightarrow2^-}\big(\text{x}-[\text{x}]\big)$
$\lim\limits_{\text{x}\rightarrow2^-}{\text{x}}-\lim\limits_{\text{x}\rightarrow2^-}[\text{x}]$
$=2-1=1$ $\Big[\because\lim\limits_{\text{x}\rightarrow\text{k}^-}[\text{x}]=\text{k}-1\Big]$
$\lim\limits_{\text{x}\rightarrow2^-}{\text{f(x)}}=\lim\limits_{\text{x}\rightarrow2^+}(3\text{x}-5)$ $[\because\text{x}>2]$
$=3(2)-5$
$=6-5$
$=1$
Thus, $\lim\limits_{\text{x}\rightarrow2^-}{\text{f(x)}}=1=\lim\limits_{\text{x}\rightarrow2^+}\text{f(x)}$
$\Rightarrow\lim\limits_{\text{x}\rightarrow2}{\text{f(x)}}=1$

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