<s>[INST] Limit $\lim_{x\to \infty} \frac{\sqrt{x+1}+\sqrt{x+2}-\sqrt{4x+12}}{\sqrt{x+2}+\sqrt{x}-\sqrt{4x+12}}$ Using the equivalency, I got the answer equal to $1$, however the answer is given as $\frac34$. 
WolframAlpha also gives $\frac34$.
Could someone tell me that the given answer is wrong or I made a mistake!
 [/INST] HINT:
$$\sqrt{4x+12}=2\sqrt{x+3}$$
Now set $1/x=h\implies h\to0^+$ 
$\sqrt{x+a}-\sqrt{x+b}$
$=\dfrac{\sqrt{1+ah}-\sqrt{1+bh}}{\sqrt h}$
$=\dfrac{1+ah-(1+bh)}{\sqrt h(\sqrt{1+ah}+\sqrt{1+bh})}$
$$\implies\dfrac{\sqrt{x+1}-\sqrt{x+3}+\sqrt{x+2}-\sqrt{x+3}}{\sqrt x-\sqrt{x+3}+\sqrt{x+2}-\sqrt{x+3}}=\dfrac{\dfrac{1-3}{\sqrt{1+h}+\sqrt{1+3h}}+\dfrac{2-3}{\sqrt{1+2h}+\sqrt{1+3h}}}{\dfrac{0-3}{\sqrt{1}+\sqrt{1+3h}}+\dfrac{2-3}{\sqrt{1+2h}+\sqrt{1+3h}}}$$
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