# solve for $\lim_{n \to \infty} \frac{6(n+1)(6n-1)!}{(6n+5)!}$

I am having trouble solving for this limit with factorials.

$$\lim_{n \to \infty} \frac{6(n+1)(6n-1)!}{(6n+5)!}$$

Any hints or suggestions would be great

• Please read this tutorial on how to typeset mathematics on this site. – N. F. Taussig Apr 30 '16 at 14:34
• On simplification, the numerator is a linear polynomial in $n$ while the denominator is a sextic polynomial in $n$. Obviously the limit as $n\to\infty$ is $0$ since $x^m\in o(x^n)\iff m\lt n$ where $o(\cdot)$ denotes the little-O notation. – learner Apr 30 '16 at 14:37

Hint: $(6n + 5)! = (6n + 5)(6n + 4)(6n + 3)(6n + 2)(6n + 1)(6n)(6n - 1)!$