Find the limit. (If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.)

 lim *x*→∞ 

9 − 

*x*

3 + 

*x*
 

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

Find the limit. (If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.) lim x→∞ (9 − x / (3 + x))

Learn how to find the limit of the function 9 - x / (3 + x) as x approaches infinity. This problem demonstrates key concepts in limits and asymptotic behavior. Suitable for students in Grades 10-12.

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