Let f(x)= x+5, & if x> 0 -4, &if x < 0 . Prove that _ x 0 f(x) do… - Vidyadip

Question

Let $\text{f(x)}=\begin{cases}\text{x}+5, & \text{if x}> 0\\\text{x}-4, &\text{if x} < 0\end{cases}.$ Prove that $\lim\limits_{\text{x}\rightarrow0}\text{f(x)}$ does not exist.

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Answer

$\text{L.H.L}=\lim\limits_{\text{x}\rightarrow0^-}\text{f(x)}$ $=\lim\limits_{\text{h}\rightarrow0}\text{f}(0-\text{h})$ $\lim\limits_{\text{h}\rightarrow0}-\text{h}-4$ $=0-4$ $=-4$ $\text{R.H.L}=\lim\limits_{\text{x}\rightarrow0^+}\text{f(x)}$ $=\lim\limits_{\text{h}\rightarrow0}\text{f}(0+\text{h})$ $=\lim\limits_{\text{h}\rightarrow0}\text{h}+5$ $=0+5$ $=5$ $\therefore\ \lim\limits_{\text{x}\rightarrow0^-}\text{f(x)}\neq\lim\limits_{\text{x}\rightarrow0^+}\text{f(x)}$ Hence $\lim\limits_{\text{x}\rightarrow0}\text{f(x)}$ does not exist.

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