Evaluate the limit of $\lim_{x \to \infty} \sqrt{x} (\sqrt {x} - \sqrt {x -a})$

Question

The question is in my book and I searched for it and got this solution. I have reached upto $$\lim_{x \to \infty} \sqrt{x} \frac { a } {(\sqrt{x} + \sqrt{x−a})}$$ However, I couldn't complete further. I don't understand the fourth and fifth step of that solution of link.

Answer 1

$${\sqrt{x}}({\sqrt{x} - \sqrt{x-a}}) = \sqrt{x}({\sqrt{x} - \sqrt{x-a}})\frac{({\sqrt{x} + \sqrt{x-a}})}{({\sqrt{x} + \sqrt{x-a}})}$$ $$= \sqrt{x}\frac{x-(x-a)}{({\sqrt{x} + \sqrt{x-a}})} = \frac{a}{\frac{({\sqrt{x} + \sqrt{x-a}})}{\sqrt{x}}} = \frac{a}{1 + \sqrt{1-\frac{a}{x}}}.$$

So, $$\lim_{x\rightarrow \infty} \sqrt{x}({\sqrt{x} - \sqrt{x-a}}) = \lim_{x\rightarrow \infty}\frac{a}{1 + \sqrt{1-\frac{a}{x}}} = \frac{a}{2}.$$

The key is to note that $$\sqrt{x} = \frac{1}{\frac{1}{\sqrt{x}}},$$ for $x \neq 0$.