If f(x)= 3x x ,& when x 0 1,&when x=0 Find whether f(x) is continuous at x = 0.

Given, f(x)= x^2-1 x-1 , if x 1 f(x)=2, if x=1 We observe (LHL at x = 1) _ x 1^- f (x)= _ x 0 (1-h) _ x 0 (1-h)= _ x 0 (1-h)^2-1 (1-h)^2-1 _ x 0 1-h^2-2h-1 1-h-1 _ x 0 h^2-2h -h _ x 0 2-h = 2 (RHL at x=1) _ x 1^+ f(x)= _ h 0 (1+h) _ h 0 (1-h)= _ h 0 (1+h)^2-1 (1+h)-1 _ h 0 1+h^2+2h-1 1+h-1 _ h 0 h^2+2h h _ h 0 h+2 =2 Also f(x) = 2 _ x 1^- f(x)= _ x 1^+ f(x)=f(1) Hence f(x) is continuous at x = 1.

If f(x)= 3x x ,& when x 0 1,&when x=0 Find whether f(x) is contin…

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Question

If $\text{f}\text{(x)}=\begin{cases}\frac{\sin3\text{x}}{\text{x}},& \text{when}\text{ x}\neq0 \\1,&\text{when} \text{ x}=0\end{cases}$ Find whether f(x) is continuous at x = 0.

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Answer

Given,
$\text{f}\text{(x)}=\frac{\text{x}^2-1}{\text{x}-1},\text{ if}\text{ x}\neq1$
$\text{f}\text{(x)}=2,\text{ if}\text{ x}=1$
We observe
$\text{(LHL at x = 1)}$
$\lim\limits_{\text{x} \rightarrow 1^-}\text{f}\text{ (x)}=\lim\limits_{\text{x} \rightarrow 0}(1-\text{h})$
$\lim\limits_{\text{x} \rightarrow 0}(1-\text{h})=\lim\limits_{\text{x} \rightarrow 0}\frac{(1-\text{h})^2-1}{(1-\text{h})^2-1}$
$\lim\limits_{\text{x} \rightarrow 0}\frac{1-\text{h}^2-2\text{h}-1}{1-\text{h}-1}$
$\lim\limits_{\text{x} \rightarrow 0}\frac{\text{h}^2-2\text{h}}{-\text{h}}$
$\lim\limits_{\text{x} \rightarrow 0}2-\text{h}$
$= 2$
$(\text{RHL at x}=1)$
$\lim\limits_{\text{x} \rightarrow 1^+}\text{f}\text{(x)}=\lim\limits_{\text{h} \rightarrow 0}(1+\text{h)}$
$\lim\limits_{\text{h} \rightarrow 0}(1-\text{h)}=\lim\limits_{\text{h} \rightarrow 0}\frac{(1+\text{h})^2-1}{(1+\text{h})-1}$
$\lim\limits_{\text{h} \rightarrow 0}\frac{1+\text{h}^2+2\text{h}-1}{1+\text{h}-1}$
$\lim\limits_{\text{h} \rightarrow 0}\frac{\text{h}^2+2\text{h}}{\text{h}}$
$\lim\limits_{\text{h} \rightarrow 0}\text{h}+2$
$=2$
Also f(x) = 2
$\lim\limits_{\text{x} \rightarrow 1^-}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow 1^+}\text{f}\text{(x)}=\text{f}(1)$
Hence f(x) is continuous at x = 1.

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