Limit of a sequence defined as a definite integral Given two sequences $$a_n=\int_{0}^{1}(1-x^2)^ndx$$ and $$b_n=\int_{0}^{1}(1-x^3)^ndx$$ where $n$ is a natural number, then find the value of $$\lim_{n \to \infty}(10\sqrt[n]{a_n}+5\sqrt[n]b_n)$$
I have no starts. Looks good though. Some hints please. Thanks.

In the same spirit as previous answers, using the gamma function, we should get $$a_n=\frac{\sqrt{\pi }}2\frac{ \Gamma (n+1)}{ \Gamma \left(n+\frac{3}{2}\right)}$$ $$b_n=\Gamma \left(\frac{4}{3}\right)\frac{ \Gamma (n+1)}{\Gamma
   \left(n+\frac{4}{3}\right)}$$ Taking logarithms and using Stirling approximation $$\log(a_n)=\left(\frac{1}{2} \log \left(\frac{1}{n}\right)+\log \left(\frac{\sqrt{\pi
   }}{2}\right)\right)+O\left(\frac{1}{n^{2}}\right)$$ which makes $$\sqrt[n]{a_n}=1+\frac{\frac{1}{2} \log \left(\frac{1}{n}\right)+\log \left(\frac{\sqrt{\pi
   }}{2}\right)}{n}+O\left(\frac{1}{n^{2}}\right)$$ Similarly $$\log(b_n)=\left(\frac{1}{3} \log \left(\frac{1}{n}\right)+\log \left(\Gamma
   \left(\frac{4}{3}\right)\right)\right)+O\left(\frac{1}{n^{2}}\right)$$ which makes $$\sqrt[n]{b_n}=1+\frac{\frac{1}{3} \log \left(\frac{1}{n}\right)+\log \left(\Gamma
   \left(\frac{4}{3}\right)\right)}{n}+O\left(\frac{1}{n^{2}}\right)$$ So, $$\alpha \sqrt[n]{a_n}+\beta \sqrt[n]{b_n}=(\alpha +\beta )+\frac{ \alpha  \log \left(\frac{\sqrt \pi }{2}\right)+\beta  \log
   \left(\Gamma \left(\frac{4}{3}\right)\right)}{n}-\frac{(3 \alpha +2 \beta ) \log (n)}{6 n}+O\left(\frac{1}{n^{2}}\right)$$ which shows the limit and how it is approached.
