To find the limit of the given expression as $x \to \infty$, the function is:

$\lim_{x \to \infty} \left( 9 - \frac{x}{3 + x} \right)$

Step-by-Step Solution:

1. Rewrite the expression:

$\lim_{x \to \infty} \left( 9 - \frac{x}{3 + x} \right)$

We want to analyze the behavior of $\frac{x}{3 + x}$ as $x \to \infty$.

2. Divide the numerator and denominator by $x$:

$\frac{x}{3 + x} = \frac{x}{x \left( \frac{3}{x} + 1 \right)} = \frac{1}{\frac{3}{x} + 1}$

As $x \to \infty$, the term $\frac{3}{x} \to 0$. So, the expression simplifies to:

$\frac{1}{\frac{3}{x} + 1} \to \frac{1}{1} = 1$

3. Substitute back into the original limit:

Now