Question

please solve it quickly...​

Answer 1

Answer:

$\displaystyle{\implies{\dfrac{2}{3}\sqrt a}}$

Step-by-step explanation:

Given :

$\displaystyle{ \lim_{x \to a}\dfrac{x-a}{x^{3/2}-a^{3/2}}}$

Using standard limit i.e.

$\displaystyle{ \lim_{x \to a}\dfrac{x^m-a^m}{x^{n}-a^{n}}=\dfrac{m}{n}a^{m-n}}$

Now ,

$\displaystyle{ \lim_{x \to a}\dfrac{x-a}{x^{3/2}-a^{3/2}}}\\\\\\\displaystyle{\dfrac{1}{\lim_{x \to a}\dfrac{x-a}{x^{3/2}-a^{3/2}}}}\\\\\\\displaystyle{\dfrac{1}{\lim_{x \to a}\dfrac{x^1-a^1}{x^{3/2}-a^{3/2}}}}\\\\\\\displaystyle{\implies\dfrac{1}{\dfrac{x^{3/2}-a^{3/2}}{x^1-a^1}}}\\\\\\\displaystyle{\implies\dfrac{1}{\dfrac{3}{2}\times\left(a^{1-3/2\right)}}}\\\\\\\displaystyle{\implies\dfrac{1}{\dfrac{3}{2}\times\left(a^{-1/2\right)}}}$

$\displaystyle{\implies{\dfrac{2}{3}\times\left(a^{1/2\right)}}}\\\\\\\displaystyle{\implies{\dfrac{2}{3}\sqrt a}}$

Thus we get answer.

Answer 2

Answer:

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

Step-by-step explanation:

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

$We \:know \: that ,\\\boxed {\lim_{x\to a }\frac{x^{m}-a^{m}}{x^n-a^{n}}=\frac{m}{n}a^{(m-n)}}$

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

$Here , m = 1 , n = \frac{3}{2}$

$= \frac{1}{\frac{3}{2}}\times a^{1-\frac{3}{2}}\\=\frac{2}{3} a^{\frac{2-3}{2}}\\=\frac{2}{3} a^{\frac{-1}{2}}\\=\frac{2}{3}\times \frac{1}{a^{\frac{1}{2}}}\\=\frac{2}{3\sqrt{a}}$

Therefore,

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

•••♪