Question

lim x -->a
x√x-a√a/x-a​

Answer 1

$\large\underline{\sf{Solution-}}$

$\: \: \: \displaystyle\sf \:\lim_{x\to a}\dfrac{x \sqrt{x} - a \sqrt{a} }{x - a}$

$\sf \: = \: \: \: \dfrac{a \sqrt{a} - a \sqrt{a} }{a - a}$

$\sf \: = \: \: \: \dfrac{0}{0}$

which is indeterminant form.

So, we can rewrite this as

$= \: \: \: \displaystyle\sf \:\lim_{x\to a}\dfrac{ {x}^{ \frac{3}{2} } - {a}^{ \frac{3}{2} } }{x - a}$

We know that,

$\boxed{ \sf \: \: \: \: \displaystyle\sf \:\lim_{x\to a}\dfrac{ {x}^{n} - {a}^{n} }{x - a} = {na}^{n - 1} }$

Now,

$\rm :\longmapsto\:Here \: n \: = \: \dfrac{3}{2}$

So, we get

$\implies\: \: \: \displaystyle\sf \:\lim_{x\to a}\dfrac{ {x}^{ \frac{3}{2} } - {a}^{ \frac{3}{2} } }{x - a} = \dfrac{3}{2} \: {a}^{ \frac{3}{2} - 1} = \dfrac{3}{2} \sqrt{a}$

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Additional Information :-

$\: \: \: \displaystyle\sf \:\lim_{x\to 0}\dfrac{sinx}{x} = 1$

$\: \: \: \displaystyle\sf \:\lim_{x\to 0}\dfrac{tanx}{x} = 1$

$\: \: \: \displaystyle\sf \:\lim_{x\to 0}\dfrac{ log(1 + x) }{x} = 1$

$\: \: \: \displaystyle\sf \:\lim_{x\to 0}\dfrac{ {e}^{x} - 1 }{x} = 1$

$\: \: \: \displaystyle\sf \:\lim_{x\to 0}\dfrac{ {a}^{x} - 1}{x} = {a}^{x} log(a)$