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Question Number 66518 by Masumsiddiqui399@gmail.com last updated on 16/Aug/19

Commented by mathmax by abdo last updated on 16/Aug/19
$let f(x)=((x(√x)−a(√a))/(x−a)) cha7gement x−a=t give lim_(x→a) f(x) =lim_(t→0) (((t+a)(√(t+a))−a(√a))/t) =lim_(t→0) (((t+a)^3 −a^3 )/(t((t+a)(√(t+a)) +a(√a)))) =lim_(t→0) ((t^3 +3t^2 a +3ta^2 )/(t{(t+a)(√(t+a))+a(√a)))) =lim_(t→0) ((t^2 +3at +3a^2 )/((t+a)(√(t+a)) +a(√a))) =((3a^2 )/(2a(√a))) =(3/2)(√a)$
$l e t f (x) = \frac{x \sqrt{x} - a \sqrt{a}}{x - a} c h a 7 g e m e n t x - a = t g i v e$ ${l i m}_{x \to a} f (x) = {l i m}_{t \to 0} \frac{(t + a) \sqrt{t + a} - a \sqrt{a}}{t}$ $= {l i m}_{t \to 0} \frac{{(t + a)}^{3} - a^{3}}{t ((t + a) \sqrt{t + a} + a \sqrt{a})}$ $= {l i m}_{t \to 0} \frac{t^{3} + 3 t^{2} a + 3 {t a}^{2}}{t {(t + a) \sqrt{t + a} + a \sqrt{a})} = {l i m}_{t \to 0} \frac{t^{2} + 3 a t + 3 a^{2}}{(t + a) \sqrt{t + a} + a \sqrt{a}}$ $= \frac{3 a^{2}}{2 a \sqrt{a}} = \frac{3}{2} \sqrt{a}$
Commented by Tony Lin last updated on 16/Aug/19

$\lim_{x \to a} \frac{x \sqrt{x} - a \sqrt{a}}{x - a}$ $= \lim_{x \to a} \frac{x^{\frac{3}{2}} - a^{\frac{3}{2}}}{x - a}$ $= \lim_{x \to a} \frac{\frac{3}{2} x^{\frac{1}{2}}}{1}$ $= \frac{3}{2} \sqrt{a}$
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