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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits2 Marks
Question
Find $\lim\limits_{\text{x}\rightarrow1}\text{f(x)}$ if $\text{f(x)}=\begin{cases}\text{x}^2-1, & \text{x} \le1\\-\text{x}^2-1, &\text{x} > 1\end{cases}.$
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Answer

$\lim\limits_{\text{x}\rightarrow1^+}\text{f(x)}=\lim\limits_{\text{x}\rightarrow1^-}\text{x}^2-1=\lim\limits_{\text{h}\rightarrow0}(-1-\text{h})^2-1=1-1=0$ $\lim\limits_{\text{x}\rightarrow1^+}\text{f(x)}=\lim\limits_{\text{x}\rightarrow1^+}\text{x}^2-\text{x}^2-1=\lim\limits_{\text{h}\rightarrow0}-(1-\text{h})^2-1=-2$ Since, $\lim\limits_{\text{x}\rightarrow1^-}\text{f(x)}\ne\lim\limits_{\text{x}\rightarrow1^+}\text{f(x)}$ $\therefore\ \lim\limits_{\text{x}\rightarrow1}\text{f(x)}$ doesnot exist.

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