Show that the function `f(x)` given by `f(x)={(xsin(1/x),x!=0),(0,x=0):}` is continuous at x = 0

Answer

Proved

Explanation

`f(x)=x \sin (\frac{1}{x})` for `x \ne 0`

and `f(x)=0` for `x=0`

For a continuous function `\LHL_{x \rightarrow a^{-}} = \RHL_{x \rightarrow a^{+}} = f(a)`

Let LHL,


`=\lim_{x \rightarrow 0^{-}} x \sin \frac{1}{x}`

`=\lim_{n \rightarrow 0}(0-4) \sin (\frac{1}{-n})`

`\rightarrow \lim_{n \rightarrow 0}-n \sin (\frac{-1}{n})`

`=\lim_{n \rightarrow 0} n \sin \frac{1} {n} `

...