Fx- x-x- if x-0- -1- if x - 0 -

Here, f(x) = {x/|x|,if x lt; 0 1, if x≥ 0 . LHL= x→ 0^ lim f(x) = x→ 0^ lim x/|x| Putting x=0 h as x→ 0^ .when x→ 0 h→ 0lim 0 h/|0 h| = h→ 0lim h/|h|= 1 ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

fx=[ x/|x|, i...

Question

f(x) = ${\begin{matrix} \frac{x}{| x |}, i f x < 0 - 1, i f x \geq 0 \end{matrix}$

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Solution

Here, f(x) = ${\begin{matrix} \frac{x}{| x |}, i f x < 0 - 1, i f x \geq 0 \end{matrix}$

LHL= $l i m x \to 0^{-}$ f(x) = $l i m x \to 0^{-}$ $\frac{x}{| x |}$

Putting x=0-h as $x \to 0^{-}$ .when $x \to 0$

$l i m h \to 0$ $\frac{0 - h}{| 0 - h |}$ = $l i m h \to 0$ $\frac{- h}{| h |} = - 1$ $∴ | - h | = h$

RHL = $l i m h \to 0^{+}$ f(x)=-1

Also, f(0)=-1 LHL=RHL=f(0)

Thus, f(x) is continuous at x=0 There is no point of discontinuity for this functions f(x).

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f(x) = {x|x|,if x <0−1, if x≥0

f(x) = ${\begin{matrix} \frac{x}{| x |}, i f x < 0 - 1, i f x \geq 0 \end{matrix}$