Find which of the function: f(x)= 3x+5,&if x 2 ^2,&if x<2 at x = … - Vidyadip

Question

Find which of the function:
$\text{f(x)}=\begin{cases}3\text{x}+5,&\text{if x}\geq2\\\text{x}^2,&\text{if x}<2\end{cases}$
at x = 2

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Answer

We have, $\text{f(x)}=\begin{cases}3\text{x}+5,&\text{if x}\geq2\\\text{x}^2,&\text{if x}<2\end{cases}$
At x = 2, $\text{L.H.L}=\lim\limits_{\text{x}\rightarrow2^-}(\text{x})^2$
$=\lim\limits_{\text{h}\rightarrow0}(2-\text{h}^2)=\lim\limits_{\text{h}\rightarrow0}(4+\text{h}^2-4\text{h})=4$
And $\text{R.H.L}=\lim\limits_{\text{x}\rightarrow2^+}(3\text{x}+5)$
$=\lim\limits_{\text{h}\rightarrow0}\big[3(2+\text{h})+5\big]=11$
Since, L.H.L ≠ R.H.L at x = 2
So, f(x) is discontinuous at x = 2.

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