Show that lim x - 0x-x- does not exist-

LHL =lim_x → 0x/|x| lim_x → 0 ( x/ x ) [ ∵ as x → 0^ , x lt; 0 slightly; ⇒ |x|= x ] = 1 RHL =lim_x → 0(x)/|x| lim_x → 0 ( x/x ) [ ∵ as x → 0^+,x gt; 0 ...

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Standard XII

Mathematics

Existence of Limit

Show that lim...

Question

Show that ${lim}_{x \to 0} \frac{x}{| x |}$ does not exist.

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Solution

$L H L = {lim}_{x \to 0} \frac{x}{| x |}$

${lim}_{x \to 0} (\frac{x}{- x})$

$[\begin{matrix} ∵ a s x \to 0^{-}, x < 0 s l i g h t l y \Rightarrow | x | = - x \end{matrix}]$

=-1

$R H L = {lim}_{x \to 0} \frac{(x)}{| x |}$

${lim}_{x \to 0} (\frac{x}{x})$

$[\begin{matrix} ∵ a s x \to 0^{+}, x > 0 s l i g h t l y \Rightarrow | x | = x \end{matrix}]$

=1

Since, $L H L \neq R H L$

$\Rightarrow {lim}_{x \to 0} \frac{x}{| x |}$ does not exist.

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Show that limx→0x|x| does not exist.

LHL=limx→0x|x| limx→0(x−x) [∵asx→0−,x<0slightly⇒|x|=−x] =-1 RHL=limx→0(x)|x| limx→0(xx) [∵asx→0+,x>0slightly⇒|x|=x] =1 Since, LHL≠RHL ⇒limx→0x|x| does not exist.

Show that ${lim}_{x \to 0} \frac{x}{| x |}$ does not exist.