Answer to Question #3811 in Calculus for lalal

Question

Find the limit of the sequence& &

1-& & & & Lim& (( 4 + 16n [sup]6[/sup]) / (7+ n [sup]6[/sup]))^1/4& &
& & & & & & & & & N ->& ∞

2-& & & Lim& (n[sup]2[/sup] + 3n ) /& e[sup]2n[/sup]&
& & & & & & & & & N ->& ∞

3-& & & Lim& (( 4 - n[sup]5[/sup]) / (n[sup]5[/sup] – 7 ) )[sup]1/3[/sup]& &
& & & & & & & & & N -> ∞

4-& & & Lim& ( 1 + 5n ) [sup]1/ n^2[/sup]
& & & & & & & & & N -> ∞

Answer 1

<img src="/cgi-bin/mimetex.cgi?%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cleft%20%28%5Cfrac%7B4%20+%2016n%5E6%7D%7B7+%20n%5E6%7D%20%5Cright%20%29%5E%7B1/4%7D%20%5Cto%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cleft%20%28%5Cfrac%7B16n%5E6%7D%7Bn%5E6%7D%20%5Cright%20%29%5E%7B1/4%7D%20%5Cto%2016%5E%7B1/4%7D%20=%202%20%5C%5C%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7Bn%5E2%20+%203n%7D%7Be%5E%7B2n%7D%7D%20%5Cto%20%5Cfrac%7B2%7D%7B4e%5E%7B2n%7D%7D%20%5Cto%200%20%5C%5C%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cleft%20%28%5Cfrac%7B4%20-n%5E5%7D%7Bn%5E5%20-7%7D%20%5Cright%20%29%5E%7B1/3%7D%20%5Cto%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cleft%20%28%5Cfrac%7B-n%5E5%7D%7Bn%5E5%7D%20%5Cright%20%29%5E%7B1/3%7D%20%5Cto%20%28-1%29%5E%7B1/3%7D%20%5C%5C%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%281%20+%205n%29%5E%7B1/n%5E2%7D%20%5Cto%201%20%5C%5C" title="\lim_{n \to \infty} \left (\frac{4 + 16n^6}{7+ n^6} \right )^{1/4} \to \lim_{n \to \infty} \left (\frac{16n^6}{n^6} \right )^{1/4} \to 16^{1/4} = 2 \\ \lim_{n \to \infty} \frac{n^2 + 3n}{e^{2n}} \to \frac{2}{4e^{2n}} \to 0 \\ \lim_{n \to \infty} \left (\frac{4 -n^5}{n^5 -7} \right )^{1/3} \to \lim_{n \to \infty} \left (\frac{-n^5}{n^5} \right )^{1/3} \to (-1)^{1/3} \\ \lim_{n \to \infty}(1 + 5n)^{1/n^2} \to 1 \\">