$$\lim_{n\to\infty}\frac{n}{1+2\sqrt{n}}$$
Given my understanding of how to solve these problems, I need to take the highest power of $n$ in the denominator and then divide both the numerator and denominator by that power. But the highest power in the denominator is lower than the one in the numerator, so I will still be left with $n^{1/2}$ in the numerator, which gives $\frac{\infty}{2}=\infty$. Is that the correct answer or am I missing something? I would have thought that since the power of the denominator is smaller than the numerator that you could factor it by long division, but I'm not sure I know how to do that.
Best Answer
It's correct, but I'd avoid writing things like $\dfrac{\infty}{2}=\infty$ for anything other than intuition.
You could use long division if you really wanted to: $$\dfrac{n}{1+2\sqrt{n}} = \dfrac{1}{2}\sqrt{n} + \dfrac{1}{4} - \frac{1}{4} \cdot \dfrac{1}{1+2\sqrt{n}}$$ This clearly gives you the same answer.