Lim_x →5/2[x] LHL: lim_x →5 /2[x] Let x=5/2 h, ⇒ h =5/2 x as x →5^ /2⇒ x lt; 5/2slightly ⇒5/2 x gt; 0 ⇒ h gt; 0 ⇒ h → 0^+ lim_h → 0 [ 5/2 h ]=2 RHL: ...

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Byju's Answer

Standard XII

Mathematics

Limit

Find lim x →-...

Question

Find ${lim}_{x \to - \frac{5}{2}} [x] .$

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Solution

${lim}_{x \to \frac{5}{2}} [x]$

$L H L : {lim}_{x \to \frac{5 -}{2}} [x]$

Let $x = \frac{5}{2} - h,$ \

$\Rightarrow h = \frac{5}{2} - x$

as $x \to \frac{5^{-}}{2} \Rightarrow x < \frac{5}{2} s l i g h t l y$

$\Rightarrow \frac{5}{2} - x > 0 \Rightarrow h > 0$

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

${lim}_{h \to 0} [\frac{5}{2} - h] = 2$

$R H L : {lim}_{x \to \frac{5 +}{2}} [x]$

Let $x = \frac{5}{2} + h,$

$\Rightarrow h = x - \frac{5}{2}$

as $x \to \frac{5^{+}}{2} \Rightarrow x > \frac{5}{2} s l i g h t l y$

$\Rightarrow x - \frac{5}{2} > 0 \Rightarrow h > 0$

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

${lim}_{h \to 0^{+}} [\frac{5}{2} + h] = 2$

We know:

Since, $= {lim}_{x \to \frac{5 -}{2}} [x] = {lim}_{x \to \frac{5 +}{2}} [x]$

$∴ {lim}_{x \to \frac{5}{2}} [x] = 2$

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Find limx→−52[x].

Find ${lim}_{x \to - \frac{5}{2}} [x] .$