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We are given the function \( r(x) = \frac{5x}{x - 8} \) and are asked to determine its behavior near the vertical asymptote \( x = 8 \). Specifically, we are looking for the one-sided limits as \( x \) approaches 8 from the left (\( x \to 8^- \)) and from the right (\( x \to 8^+ \)).

### Step-by-step analysis:

1. **For \( x \to 8^- \) (approaching 8 from the left):**
   - As \( x \) approaches 8 from the left, \( x - 8 \) becomes a small negative number (since values of \( x \) are slightly less than 8).
   - Therefore, the denominator \( x - 8 \) will approach \( 0^- \) (a small negative number).
   - The numerator \( 5x \) will approach \( 5(8) = 40 \), which is positive.
   - So, the whole fraction approaches \( \frac{40}{0^-} \), which tends to \( -\infty \).

   Thus, \( \lim_{x \to 8^-} r(x) = -\infty \).

2. **For \( x \to 8^+ \) (approaching 8 from the right):**
   - As \( x \) approaches 8 from the right, \( x - 8 \) becomes a small positive number (since values of \( x \) are slightly more than 8).
   - Therefore, the denominator \( x - 8 \) will approach \( 0^+ \) (a small positive number).
   - The numerator \( 5x \) will approach \( 40 \), which is positive.
   - So, the whole fraction approaches \( \frac{40}{0^+} \), which tends to \( +\infty \).

   Thus, \( \lim_{x \to 8^+} r(x) = +\infty \).

### Conclusion:
The correct option describes the behavior as:
\[
\lim_{x \to 8^-} r(x) = -\infty \quad \text{and} \quad \lim_{x \to 8^+} r(x) = +\infty.
\]
This corresponds to the **last option** in the image.

Would you like a detailed explanation of why vertical asymptotes behave this way? Here are five related questions to explore this topic further:

1. How do you identify the location of vertical asymptotes in a rational function?
2. What happens to the graph of a function near its vertical asymptote?
3. Can a function have more than one vertical asymptote?
4. What is the difference between vertical and horizontal asymptotes?
5. How do you calculate limits involving infinity for more complex rational functions?

**Tip:** Always remember that vertical asymptotes occur where the denominator approaches zero but the numerator does not.

Which of the following describes the behavior of the function r(x) = 5x / (x - 8) near its asymptote?

Explore how the rational function r(x) = 5x / (x - 8) behaves near its vertical asymptote x = 8. Learn the step-by-step solution using limit theorems and identify how the function approaches infinity from both sides of the asymptote. Suitable for high school students in Grades 11-12.

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