calculus - Find the value of : $lim_{xtoinfty}frac{sqrt{x-1} - sqrt{x-2}}{sqrt{x-2} - sqrt{x-3}}$

I'm trying to solve evaluate this limit

$$\lim_{x\to\infty}\frac{\sqrt{x-1} - \sqrt{x-2}}{\sqrt{x-2} - \sqrt{x-3}}.$$

I've tried to rationalize the denominator but this is what I've got

$$\lim_{x\to\infty}(\sqrt{x-1} - \sqrt{x-2})({\sqrt{x-2} + \sqrt{x-3}})$$

and I don't know how to remove these indeterminate forms $(\infty - \infty)$.

EDIT: without l'Hospital's rule (if possible).

Answer

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As $\;x\to\infty\;$ we can assume $\;x>0\;$ , so:

$$\frac{\sqrt{x-1}-\sqrt{x-2}}{\sqrt{x-2}-\sqrt{x-3}}=\frac{\sqrt{x-2}+\sqrt{x-3}}{\sqrt{x-1}+\sqrt{x-2}}=\frac{\sqrt{1-\frac2x}+\sqrt{1-\frac3x}}{\sqrt{1-\frac1x}+\sqrt{1-\frac2x}}\xrightarrow[x\to\infty]{}1$$

Further hint: the first step was multiplying by conjugate of both the numerator and the denominator.