We are given the function:

$f(x) = \frac{x^2 - 81}{x + 9}$

and asked to find the conjecture about the limits and sketch the graph for $a = -9$.

Step 1: Simplify the function

First, simplify $f(x)$:

$f(x) = \frac{x^2 - 81}{x + 9}$

Notice that $x^2 - 81$ is a difference of squares:

$x^2 - 81 = (x + 9)(x - 9)$

So,

$f(x) = \frac{(x + 9)(x - 9)}{x + 9}$

For $x \neq -9$, we can cancel out $(x + 9)$, leaving:

$f(x) = x - 9 \quad \text{for} \quad x \neq -9$

Step 2: Analyze the behavior at $a = -9$

The function $f(x) = x - 9$ is a linear function everywhere except at $x = -9$. At $x = -9$, there is a hole (discontinuity) in the graph because the original function had a denominator of $x + 9$, which would make the function undefined at that point.

Now, let’s calculate the limits:

1. Left-hand limit $\lim_{x \to -9^-} f(x)$: As $x$ approaches $-9$ from the left, $f(x)$ behaves as $f(x) = x - 9$. Thus:

   $\lim_{x \to -9^-} f(x) = -9 - 9 = -18$

2. Right-hand limit $\lim_{x \to -9^+} f(x)$: As $x$ approaches $-9$ from the right, $f(x)$ behaves similarly:

   $\lim_{x \to -9^+} f(x) = -9 - 9 = -18$

3. The value of the function at $x = -9$: Since the function is undefined at $x = -9$ (due to the denominator being zero), we cannot directly evaluate $f(-9)$.

4. Overall limit at $x = -9$: Since the left-hand and right-hand limits are equal:

   $\lim_{x \to -9} f(x) = -18$

Step 3: Graph sketch

Given the information, the graph of $f(x)$ is a line $y = x - 9$ everywhere except at $x = -9$, where there is a hole.

• The graph has a hole at $x = -9$, and at this point, $f(x)$ does not exist.
• The function approaches $-18$ as $x$ approaches $-9$ from both sides, but $f(x)$ is undefined exactly at $x = -9$.

Step 4: Conjecture about the graph and limits

• The left-hand limit: $\lim_{x \to -9^-} f(x) = -18$
• The right-hand limit: $\lim_{x \to -9^+} f(x) = -18$
• The overall limit: $\lim_{x \to -9} f(x) = -18$
• Value of the function at $x = -9$: $f(-9)$ does not exist.

Step 5: Identify the correct graph

From the options provided:

• Option A shows a graph with a hole at $(-9, 18)$, which does not match our analysis.
• Option B shows a hole at $(-9, 0)$, which is also incorrect.
• Option C shows a hole at $(-9, -18)$, which matches our conjecture.

Thus, the correct graph is Option C.

Next Steps:

1. Would you like a more detailed explanation of the graphing process for functions with holes?
2. Should I explore how to handle similar functions with discontinuities?
3. Would you like to learn about how to calculate limits for functions with holes?
4. Do you need more information on limits or continuity?
5. Would you like to investigate another example involving rational functions?

Tip: Always check for factors that can be canceled out in a rational function, as this may reveal hidden behavior at certain points.