Factorials and binomials

Limit of binomial coefficient over quadratic ●○○○○ 10 ptsCalculate $\lim_{n \to \infty} \frac{\binom{n}{2}}{n^2+1}$ . Give your answer as a decimal.
Limit with sine of factorial ●○○○○ 10 ptsEvaluate $\lim_{n\to\infty}\frac{n^{1/3}\sin(n!)}{1+n}$ .
Limit of a factorial quotient sequence ●●○○○ 20 ptsFind $\lim_{n\to\infty} a_n$ , where $a_n= \frac{(n+1)!\,(n-1)!}{(2n)!}$ .
Limit of n factorial over n to the n ●●○○○ 20 ptsFind $\lim_{n \to \infty} \frac{n!}{n^n}$ .
Limit of a ratio of factorial sums ●●○○○ 20 ptsFind $\lim_{n\to\infty}\frac{1!+2!+\cdots+n!}{n!}$ .
Limit of xth root of factorial over x ●●●○○ 50 ptsFind the value of $\lim_{n\to\infty} \frac{\sqrt[n]{n!}}{n}$ , where $n$ is a positive integer. Round to two decimal places.
Limit of nth root of factorial over n ●●●○○ 50 ptsFind $\lim_{n\to\infty} \frac{(n!)^{1/n}}{n}$ . Round to three decimal places.
Limit of a central binomial ratio ●●●○○ 50 ptsFind $\lim_{n \to \infty} \frac{(2n)!\,\sqrt{n}}{(n!)^2\,4^n}$ . Round to two decimal places.

More in Limits of sequences