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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Find which of the function:
$\text{f(x)}=\begin{cases}\frac{|\text{x}-4|}{2(\text{x}-4)},&\text{if x}\neq4\\0,&\text{if x}=4\end{cases}$
at x = 4
✓

Answer

The condition for function f to be a continuous at x = a is given by $=\lim\limits_{\text{x}\rightarrow\text{a}^-}\text{f(x)}=\lim\limits_{\text{x}\rightarrow\text{a}^+}\text{f(x)}=\text{f(a)}$
Consider, $\text{f(x)}=\begin{cases}\frac{|\text{x}-4|}{2(\text{x}-4)},&\text{if x}\neq4\\0,&\text{if x}=4\end{cases}$ at x = 4.
At x = 4, $\text{L.H.L}=\lim\limits_{\text{x}\rightarrow4^-}\frac{|\text{x}-4|}{2(\text{x}-4)}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{|4-\text{h}-4|}{2\big[(4-\text{h})-4\big]}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{|-\text{h}|}{-2\text{h}}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\text{h}}{-2\text{h}}=\frac{-1}{2}\text{ and f}(4)=0\neq\text{L.H.L}$
So, f(x) is discontinuous at x = 4.

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