Fraction!

Relevant wiki: Sum of n, n², or n³

$\begin{aligned} L & = \lim_{n \to \infty} \frac {\color{#3D99F6}1+16+81+\cdots + n^4}{{\color{#D61F06}(1+4+9+16+\cdots + n^2)}(n-7)(n+8)} & \small \color{#3D99F6} \text{Use Faulhaber's formula for }\sum_{k=1}^n k^4 \text{ (see reference)} \\ & = \lim_{n \to \infty} \frac {\color{#3D99F6}\frac 1{30}n(n+1)(2n+1)(3n^2+3n-1)}{{\color{#D61F06}\frac 16n(n+1)(2n+1)}(n-7)(n+8)} \\ & = \lim_{n \to \infty} \frac {3n^2+3n-1}{5(n-7)(n+8)} \\ & = \lim_{n \to \infty} \frac {3n^2+3n-1}{5(n^2+n-56)} & \small \color{#3D99F6} \text{Divide up and down by } n^2 \\ & = \lim_{n \to \infty} \frac {3+\frac 3n-\frac 1{n^2}}{5\left(1+\frac 1n-\frac {56}{n^2}\right)} \\ & = \frac 35 \end{aligned}$

$\implies a + 5b = 3 + 5(5) = \boxed{28}$