<s>[INST] Proving $\lim _{x\to \infty }\left(\frac{\sqrt{x+1}-\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-3}}\right) = \frac35$ $$\lim _{x\to \infty }\left(\frac{\sqrt{x+1}-\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-3}}\right)$$
Can someone help me to solve it?
result of online calculator: 3/5
 [/INST] Hint:
$$\frac{\sqrt{x+1}-\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-3}} = \frac{\sqrt{x+1}-\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-3}}\cdot\frac{\sqrt{x+1}+\sqrt{x-2}}{\sqrt{x+2}+\sqrt{x-3}}\cdot\frac{\sqrt{x+2}+\sqrt{x-3}}{\sqrt{x+1}+\sqrt{x-2}}$$
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