show steps on how to get the horizontal asymptotes: *f*(*x*)=(2x^2-5x-12)/x^2-16

To find the horizontal asymptotes of a rational function, we need to analyze the behavior of the function as \( x \) approaches infinity or negative infinity.

The function given is:

\[
f(x) = \frac{2x^2 - 5x - 12}{x^2 - 16}
\]

### Steps to Find Horizontal Asymptotes:

1. **Identify the degrees of the polynomials in the numerator and denominator.**

   The degrees of the polynomials are:
   - The numerator is \( 2x^2 - 5x - 12 \), which is a degree 2 polynomial (highest power of \( x \) is 2).
   - The denominator is \( x^2 - 16 \), which is also a degree 2 polynomial.

2. **Compare the degrees of the numerator and the denominator.**
   
   - If the degree of the numerator is **less than** the degree of the denominator, the horizontal asymptote is \( y = 0 \).
   - If the degree of the numerator is **equal to** the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
   - If the degree of the numerator is **greater than** the degree of the denominator, there is no horizontal asymptote.

   In this case, both the numerator and denominator are degree 2 polynomials.

3. **Find the ratio of the leading coefficients.**
   
   - The leading coefficient of the numerator \( 2x^2 \) is 2.
   - The leading coefficient of the denominator \( x^2 \) is 1.

   So, the horizontal asymptote is given by:

   \[
   y = \frac{2}{1} = 2
   \]

### Conclusion:
The horizontal asymptote of the function \( f(x) = \frac{2x^2 - 5x - 12}{x^2 - 16} \) is:

\[
y = 2
\]

Would you like a more detailed explanation on any step or have any questions?

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Here are 5 questions related to this topic:
1. How do vertical asymptotes differ from horizontal asymptotes?
2. Can there be more than one horizontal asymptote for a rational function?
3. What are the conditions under which a rational function has an oblique asymptote?
4. How do the degrees of polynomials affect the type of asymptotes in a function?
5. Can you find the vertical asymptotes of the function \( f(x) = \frac{2x^2 - 5x - 12}{x^2 - 16} \)?

**Tip**: When finding horizontal asymptotes, always focus on the leading terms of both the numerator and the denominator for large values of \( x \).

show steps on how to get the horizontal asymptotes: f(x) = (2x^2 - 5x - 12)/x^2 - 16

Learn how to determine the horizontal asymptote of the rational function f(x) = (2x^2 - 5x - 12)/x^2 - 16 by comparing the degrees of the numerator and denominator. This guide explains the steps to identify horizontal asymptotes using the leading coefficients. Suitable for high school students in Grades 10-12.

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