We are given the limit of a rational function as $x$ approaches infinity: $\lim_{x\to\infty} \frac{8x^2 - x + 1}{ax^2 + 8} = 1$ We need to find the value of $a$.

1. Problem Description

We are given the limit of a rational function as $x$ approaches infinity:

$\lim_{x\to\infty} \frac{8x^2 - x + 1}{ax^2 + 8} = 1$

We need to find the value of $a$.

2. Solution Steps

To find the limit as $x$ approaches infinity for a rational function, we can divide both the numerator and the denominator by the highest power of $x$ present, which in this case is $x^2$.

$\lim_{x\to\infty} \frac{8x^2 - x + 1}{ax^2 + 8} = \lim_{x\to\infty} \frac{\frac{8x^2}{x^2} - \frac{x}{x^2} + \frac{1}{x^2}}{\frac{ax^2}{x^2} + \frac{8}{x^2}} = \lim_{x\to\infty} \frac{8 - \frac{1}{x} + \frac{1}{x^2}}{a + \frac{8}{x^2}}$

As $x$ approaches infinity, $\frac{1}{x}$ and $\frac{1}{x^2}$ approach

0. Thus,

$\lim_{x\to\infty} \frac{8 - \frac{1}{x} + \frac{1}{x^2}}{a + \frac{8}{x^2}} = \frac{8 - 0 + 0}{a + 0} = \frac{8}{a}$

We are given that this limit is equal to

1. $\frac{8}{a} = 1$

Multiplying both sides by $a$, we get:

$8 = a$

3. Final Answer

$a = 8$