The limit of the ratio of ${(n+1)^{\frac 12}-(n^3+1)^{\frac 13}}$ and ${(n+1)^{\frac 14}-(n^5+1)^{\frac 15}}$

Question

Find the limit $$\lim_{n\to\infty}\dfrac{(n+1)^{\frac 12}-(n^3+1)^{\frac 13}}{(n+1)^{\frac 14}-(n^5+1)^{\frac 15}}$$

I've already tried to multiply on conjugate expression but I failed. Can you explain me the steps?

Answer 1

The limit is $1$ , since in the case of sums of powers, be they polynomial or exponential, only the predominant factor matters, which in this case is $n$ in both places, since $\sqrt[3]{n^3} = \sqrt[5]{n^5} = n$ , which is greater than either $\sqrt n$ or $\sqrt[4]n$ , so our limit becomes $\lim_{n\to\infty} \frac nn = 1$ .

Answer 2

$(n+1)^{1/2}=n^{1/2}(1+1/n)^{1/2}=n^{1/2}(1+1/(2n)+...)\\=n^{1/2}+(1/2)n^{-1/2}+...$
Similar Taylor series work for the other factors.
Should that have been $(n^2+1)^{1/2}$ instead of $(n+1)^{1/2}$?