Evaluation of this limit

Question

I am looking in a calculator how to evaluate the limit as x approaches infinity for the following function: $\sqrt{x} -\sqrt{x-1}$.

In the evaluation the calculator applies the following algebraic property: $$a+b = \frac{a^2 - b^2}{a-b}.$$ I am wondering if this algebraic property is necessary or we can solve the evaluation of the limit in a simpler way.

Anyway, the final result is zero. Could you remind the properties of limits at infinity which I need to consider in this specific case?

Answer 1

Let $x=\csc^22y$

As $x\to+\infty,y\to0^+$

$$\lim_{x\to\infty}(\sqrt x-\sqrt{x-1})=\lim_{y\to0^+}(\csc2y-\cot2y)=\lim_{y\to0^+}\dfrac{2\sin^2y}{2\sin y\cos y}$$

As $y\to0,\sin y\to0\implies\sin y\ne0$

$$\implies\lim_{y\to0^+}\dfrac{2\sin^2y}{2\sin y\cos y}=\lim_{y\to0^+}\tan y=?$$

Answer 2

$$(\sqrt{x}-\sqrt{x-1}).\frac{(\sqrt{x}+\sqrt{x-1})}{(\sqrt{x}+\sqrt{x-1})}=\frac{1}{(\sqrt{x}+\sqrt{x-1})}$$ when $x\rightarrow \infty $ the limit is $0$