Prove that lim x - a - x - a - for all a - R- Also- prove that lim x - 1- x -0

Lim_x → a^+ [x] Let x=a+h ⇒ h =x a as x → a^+ ⇒ x gt; a slightly ⇒ x a gt; 0 ⇒ h gt; 0 ⇒ h → 0^+ ⇒lim_h → 0^+[a+h]=[a] ⇒lim_h → a^+[x]=[a]∀ a ∈ R Also ...

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Standard XII

Mathematics

Limit

Prove that li...

Question

Prove that ${lim}_{x \to a^{+}} [x]$ =[a] for all $a \in R$ . Also, prove that ${lim}_{x \to 1^{-}} [x] = 0$

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Solution

${lim}_{x \to a^{+}} [x]$

Let $x = a + h \Rightarrow h = x - a$

as $x \to a^{+} \Rightarrow x > a$ slightly

$\Rightarrow x - a > 0 \Rightarrow h > 0$

$\Rightarrow h \to 0^{+}$

$\Rightarrow {lim}_{h \to 0^{+}} [a + h] = [a]$

$\Rightarrow {lim}_{h \to a^{+}} [x] = [a] \forall a \in R$

Also, ${lim}_{x \to 1^{+}} [x]$

Let x =1-h

$\Rightarrow h = 1 - x$

as $x \to 1^{-} \Rightarrow x < 1$ slightly

$\Rightarrow 1 - x < 0 \Rightarrow h < 0$

$\Rightarrow h \to 0^{-}$

$= {lim}_{h \to 0^{-}} [1 - h] = 0$

$= {lim}_{x \to 1^{-}} [x] = 0$

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Prove that limx→a+[x] =[a] for all a∈R. Also, prove that limx→1−[x]=0

limx→a+[x] Let x=a+h⇒h=x−a as x→a+⇒x>a slightly ⇒x−a>0⇒h>0 ⇒h→0+ ⇒limh→0+[a+h]=[a] ⇒limh→a+[x]=[a]∀a∈R Also, limx→1+[x] Let x =1-h ⇒h=1−x as x→1−⇒x<1 slightly ⇒1−x<0⇒h<0 ⇒h→0− =limh→0−[1−h]=0 =limx→1−[x]=0

Prove that ${lim}_{x \to a^{+}} [x]$ =[a] for all $a \in R$ . Also, prove that ${lim}_{x \to 1^{-}} [x] = 0$