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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits3 Marks
Question
Find k so that $\lim\limits_{\text{x}\rightarrow2}\text{f(x)}$ may exist, where $\text{f(x)}=\begin{cases}2\text{x}+3, & \text{x}\le 2\\\text{x}+\text{k}, & \text{x} > 2\end{cases}.$
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Answer

$\lim\limits_{\text{x}\rightarrow2^-}\text{f(x)}=\lim\limits_{\text{x}\rightarrow2^-}(2\text{x}+3)$ $=2(2)+3$ $=7$ $\therefore\lim\limits_{\text{x}\rightarrow2^-}\text{f(x)}=7$ Also, $\lim\limits_{\text{x}\rightarrow2^+}\text{f(x)}=\lim\limits_{\text{x}\rightarrow2^+}(\text{x}+\text{k})$ $=(2+\text{k})$ Since, $\lim\limits_{\text{x}\rightarrow2}\text{f(x)}$ exists (given) $\therefore\ \lim\limits_{\text{x}\rightarrow2^-}\text{f(x)}=\lim\limits_{\text{x}\rightarrow2^+}\text{f(x)}$ $\Rightarrow7=2+\text{k}$ $\Rightarrow\text{k}=5$

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