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 of a rational function as $x$ approaches infinity, we can divide both the numerator and the denominator by the highest power of $x$ that appears in the denominator. In this case, the highest power 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 $x \to \infty$, the terms $\frac{1}{x}$ and $\frac{1}{x^2}$ approach

0. Therefore,

$$$$

\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 the limit is equal to 1:

$$$$

\frac{8}{a} = 1

$$$$

Multiplying both sides by $a$, we get:

$$$$

8 = a

$$$$

Thus, $a = 8$.

3. Final Answer

$a = 8$