We are given the function:

$f(x) = \frac{x^2 - 10x + 24}{x^{12} - 6x^{11}}$

To find the vertical asymptotes:

Vertical asymptotes occur when the denominator is zero and the numerator is non-zero at the same points. Therefore, we need to find where the denominator $x^{12} - 6x^{11}$ is equal to zero.

Step 1: Factor the Denominator

We can factor the denominator $x^{12} - 6x^{11}$ as follows:

$x^{12} - 6x^{11} = x^{11}(x - 6)$

Now, set the denominator equal to zero:

$x^{11}(x - 6) = 0$

This gives us two solutions:

1. $x^{11} = 0$ → $x = 0$
2. $x - 6 = 0$ → $x = 6$

Step 2: Check the Numerator at These Points

We need to ensure the numerator is non-zero at these values of $x$.

The numerator is $x^2 - 10x + 24$.

• At $x = 0$:

$0^2 - 10(0) + 24 = 24 \quad (\text{non-zero})$

• At $x = 6$:

$6^2 - 10(6) + 24 = 36 - 60 + 24 = 0 \quad (\text{zero})$

Conclusion:

• There is a vertical asymptote at $x = 0$ since the denominator is zero and the numerator is non-zero.
• At $x = 6$, there is no vertical asymptote because both the numerator and denominator are zero, indicating a hole in the graph rather than an asymptote.

Thus, the function has a vertical asymptote at $x = 0$, and a hole at $x = 6$.

Would you like further details or have any questions?

Relative questions:

1. How do we determine horizontal asymptotes for this function?
2. What is the behavior of the function near the vertical asymptote $x = 0$?
3. How can we find the hole at $x = 6$ more precisely?
4. Can you provide more insight into why holes occur in rational functions?
5. What other techniques are available to find vertical asymptotes beyond factoring?

Tip:

Always check both the numerator and denominator for zeros when determining vertical asymptotes to identify potential holes in the graph.