Question

Вычислить предел: $\lim_{n \to \infty} \frac{\sqrt[n]{2019n^n+n+1}+\sqrt{n} }{\sqrt{n+2}+3n+1 }$

Answer 1

$\lim_{n \to \infty} \dfrac{\sqrt[n]{2019n^n+n+1}+\sqrt{n}  }{\sqrt{n+2}+3n+1 }=  \lim_{n \to \infty} \dfrac{\sqrt[n]{2019+\dfrac{1}{n^{n-1}} +\dfrac{1}{n^{n}}}+\dfrac{1}{\sqrt{n}}  }{\sqrt{\dfrac{1}{n}+\dfrac{2}{n^2}}+3+\dfrac{1}{n} }=\\ \lim_{n \to \infty} \dfrac{1+0 }{0+3+0 }=\dfrac{1}{3}$