calculus - What is $limlimits_{n rightarrow infty} frac{sqrt{n+1} + 3}{sqrt{n+2} - 4}$?

$$\begin{equation*} \lim_{n \rightarrow \infty} \frac{\sqrt{n+1} + 3}{\sqrt{n+2} - 4} \end{equation*}$$

I know that I should multiply by the conjugate if I have square roots in either the numerator or the denominator but what if I have square roots in both numerator and denominator?

Could anyone help me please?

Answer

HINT

By intuition we have that the $\sqrt n$ terms are dominant on the others and therefore in the limit

$$\frac{\sqrt{(n+1)}+ 3}{\sqrt{(n+2)} - 4}\sim \frac{\sqrt n}{\sqrt n}=1$$

To check and formalize rigoursly this fact let consider

$$\frac{\sqrt{(n+1)}+ 3}{\sqrt{(n+2)} - 4}=\frac{\sqrt n}{\sqrt n}\frac{\sqrt{(1+1/n)}+ 3/\sqrt n}{\sqrt{(1+2/n)} - 4/\sqrt n}$$

then take the limit.