Limit of a fraction involving multiple square roots

Question

$$\lim_{x\to2}{\frac{\sqrt{3x-2}-\sqrt{5x-6}}{\sqrt{2x-1}-\sqrt{x+1}}}$$

Evaluate the limit.

Thanks for any help

Answer 1

$$\lim_{x\to2}{\frac{\sqrt{3x-2}-\sqrt{5x-6}}{\sqrt{2x-1}-\sqrt{x+1}}}=\lim_{x\to2}{\frac{(3x-2-(5x-6))(\sqrt{2x-1}+\sqrt{x+1})}{(2x-1-(x+1))(\sqrt{3x-2}+\sqrt{5x-6})}}=$$ $$=-2\cdot\frac{\sqrt3+\sqrt3}{2+2}=-\sqrt3$$

Answer 2

That stinks a lot as homework. A general hint for solving such a problem is to "rationalize" both numerator and denominator i.e. multiply the whole thing by

$$\frac{\sqrt{3x-2}+\sqrt{5x-6}}{\sqrt{3x-2}+\sqrt{5x-6}}\cdot \frac{\sqrt{2x-1}+\sqrt{x+1}}{\sqrt{2x-1}+\sqrt{x+1}}$$

Answer 3

Hint:

Set $x-2=h$

and rationalize the denominator & the numerator.

Setting limit to $0$ often eases of the calculation