What is the limit of the following term?

Question

Find the limit of the following term:

$$\lim_{x\to\infty} \sqrt{2x-3}-\sqrt{ax+b},$$

while $a\in\mathbb{R}^+,b\in\mathbb{R}$.

Please, I need help!!!

Answer 1

$$\sqrt{2x-3}-\sqrt{ax+b} = \dfrac{(\sqrt{2x-3} - \sqrt{ax+b})(\sqrt{2x+3} + \sqrt{ax+b})}{\sqrt{2x-3} + \sqrt{ax+b}} =\frac{(2x-3)-(ax+b)}{\sqrt{2x-3} + \sqrt{ax+b}}$$

$$= \frac{(2-a)x-(3+b)}{\sqrt{2x-3}+\sqrt{ax+b}}$$

Can you take the limit now? The degree of the numerator exceeds that of the denominator: try dividing numerator and denominator by $\sqrt x$. Then use the fact that given $a>0$, $2-a>0$ provided $0\leq a \lt 2$, $2-a = 0$ when $a=2$, and $2-a \lt 0$ when $a>2$.

Answer 2

Hint: Multiply and divide by the conjugate.