<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] Rewrite $$A=\frac{\sqrt{x+1}-\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-3}}=\frac{\sqrt{x}\sqrt{1+1/x}-\sqrt{x}\sqrt{1-2/x}}{\sqrt{x}\sqrt{1+2/x}-\sqrt{x}\sqrt{1-3/x}}=\frac{\sqrt{1+1/x}-\sqrt{1-2/x}}{\sqrt{1+2/x}-\sqrt{1-3/x}}$$ Replace $\frac 1x$ by $y$; so $$A=\frac{\sqrt{1+y}-\sqrt{1-2y}}{\sqrt{1+2y}-\sqrt{1-3y}}$$ Now, use the fact  that, when $z$ is small compared to $1$,$\sqrt{1+z}\approx 1+\frac z2$. Replace $z$ by the appropriate value for each radical.
I am sure that you can take from here.
</s>