Limit $\lim_{x\to2^+}x{\sqrt{x-1}\over\sqrt{x-2}}$

Question

I am having problems with this limit problem, I tried to rationalize and keep getting that the limit is $∞$ but in the book the answer is $2$, am I making a mistake here ?

$$\lim_{x\to 2^+} x\frac{\sqrt{x-1}}{\sqrt{x-2}}$$

Answer 1

Note that

$$\lim_{x\to2^+}x{\sqrt{x-1}\over\sqrt{x-2}}=\left(2\cdot\frac{1}{0^+}\right)=+\infty$$

Answer 2

The limit is $+\infty$ it of the form ${2\over 0}$