Computing a limit form first principles

Question

How can you prove, via first principles, that the limit $$\lim_{h\rightarrow 0}\frac{\sqrt{(1-h)^2+(2+h)^2}-\sqrt{5}}{h}$$ exists?

Somehow, I wasn't able to do it, without using specific properties of the function $\frac{\sqrt{(1-h)^2+(2+h)^2}-\sqrt{5}}{h}$ and just using basic propositions about limits.

Answer 1

Hint Multiply the numerator and denominator by $\sqrt{(1-h)^2+(2+h)^2}+\sqrt{5}$. Then you get $$\lim_{h\to 0}\frac{(1-h)^2+(2+h)^2-5}{h(\sqrt{(1-h)^2+(2+h)^2}+\sqrt{5})}=\lim_{h\to 0}\frac{1-2h+h^2+4+4h+h^2-5}{h2\sqrt 5}\\=\lim_{h\to 0}\frac{h(2+2h)}{h2\sqrt 5}=\lim_{h\to 0}\frac{2+2h}{2\sqrt 5}=\frac 1{\sqrt 5}$$

Answer 2

What happens when you multiply the numerator and denominator by the (clearly non-zero) conjugate $$\sqrt{(1-h)^2+(2+h)^2} + \sqrt{5}?$$