Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

20. lim x^2 − 3x /x^2-9 = −2.5, −2.9, −2.95, −2.99, −2.999, −2.9999, x→−3 x2 − 9 −3.5, −3.1, −3.05, −3.01, −3.001, −3.0001

To estimate the value of the limit \( \lim_{x \to -3} \frac{x^2 - 3x}{x^2 - 9} \), we can evaluate the function at points increasingly close to \( x = -3 \) from both the left and right sides.

The function to evaluate is:

\[
f(x) = \frac{x^2 - 3x}{x^2 - 9}
\]

We need to evaluate this function at the given values of \( x \): \( -2.5 \), \( -2.9 \), \( -2.95 \), \( -2.99 \), \( -2.999 \), \( -2.9999 \), \( -3.5 \), \( -3.1 \), \( -3.05 \), \( -3.01 \), \( -3.001 \), and \( -3.0001 \).

Let me compute the values of \( f(x) \) at these points.The values of the function \( f(x) = \frac{x^2 - 3x}{x^2 - 9} \) at the given points are as follows:

- \( f(-2.5) = -5.000000 \)
- \( f(-2.9) = -29.000000 \)
- \( f(-2.95) = -59.000000 \)
- \( f(-2.99) = -299.000000 \)
- \( f(-2.999) = -2999.000000 \)
- \( f(-2.9999) = -29999.000000 \)
- \( f(-3.5) = 7.000000 \)
- \( f(-3.1) = 31.000000 \)
- \( f(-3.05) = 61.000000 \)
- \( f(-3.01) = 301.000000 \)
- \( f(-3.001) = 3001.000000 \)
- \( f(-3.0001) = 30001.000000 \)

As \( x \) approaches \( -3 \) from the left, the function values become increasingly large negative numbers. As \( x \) approaches \( -3 \) from the right, the function values become increasingly large positive numbers. This suggests that the function has a vertical asymptote at \( x = -3 \), and therefore the limit does not exist.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What is the general approach to finding the limit of a rational function?
2. How do vertical asymptotes affect the existence of a limit?
3. Can you graph the function to visualize the behavior near \( x = -3 \)?
4. What happens to the limit if we approach \( x = 3 \) instead of \( x = -3 \)?
5. How can we algebraically determine the presence of a vertical asymptote in a function?

**Tip:** When evaluating limits, always consider the behavior of the function from both the left and right sides of the point in question to identify potential asymptotes or discontinuities.

Learn how to evaluate the limit \( \lim_{x \to -3} \frac{x^2 - 3x}{x^2 - 9} \) by analyzing the function's behavior near the point \( x = -3 \). Understand the concept of vertical asymptotes and the approach from both sides to determine the limit. Suitable for advanced high school students studying calculus.

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