Let f(x)= x+1, & if x> 0 -1, &if x < 0 . Prove that _ x 0 f(x) do… - Vidyadip

Question

Let $\text{f(x)}=\begin{cases}\text{x}+1, & \text{if x}> 0\\\text{x}-1, &\text{if x} < 0\end{cases}.$ Prove that $\lim\limits_{\text{x}\rightarrow0}\text{f(x)}$ does not exist.

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Answer

$\lim\limits_{\text{x}\rightarrow0^+}\text{f(x)}=\lim\limits_{\text{x}\rightarrow0^+}\text{ x}+1$ $=\lim\limits_{\text{h}\rightarrow0}(0+\text{h})+1=1$ Also, $\lim\limits_{\text{x}\rightarrow0^+}\text{f(x)}\neq\lim\limits_{\text{x}\rightarrow0^-}\text{f(x)}$ $=\lim\limits_{\text{h}\rightarrow0}(0-\text{h})-1=-1$ $\Rightarrow\lim\limits_{\text{x}\rightarrow0^+}\text{f(x)}=\lim\limits_{\text{x}\rightarrow0^-}\text{f(x)}$ Hence, limit does not exist.

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