Find the horizontal asymptote of f(x) = 13x^2+6x/12x^2-5

To find the horizontal asymptote of the function \( f(x) = \frac{13x^2 + 6x}{12x^2 - 5} \), we compare the degrees of the polynomials in the numerator and the denominator.

### Step 1: Compare the degrees of the polynomials
- The degree of the numerator \( 13x^2 + 6x \) is 2.
- The degree of the denominator \( 12x^2 - 5 \) is 2.

Since both the numerator and denominator have the same degree (degree 2), the horizontal asymptote is determined by the ratio of the leading coefficients.

### Step 2: Ratio of the leading coefficients
The leading coefficient of the numerator is 13, and the leading coefficient of the denominator is 12.

Thus, the horizontal asymptote is:
\[
y = \frac{13}{12}
\]

So, the horizontal asymptote of the function \( f(x) \) is \( y = \frac{13}{12} \).

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Do you want more details or have any questions?  
Here are 5 related questions to expand your understanding:

1. How do you find the vertical asymptote of a rational function?
2. What happens to a rational function at its horizontal asymptote as \( x \to \infty \)?
3. What is the horizontal asymptote of a function when the degree of the denominator is higher than the numerator?
4. Can a rational function cross its horizontal asymptote?
5. How do you find the oblique asymptote of a rational function when the degree of the numerator is greater than the denominator?

**Tip:** When the degrees of the numerator and denominator of a rational function are the same, always divide their leading coefficients

Find the horizontal asymptote of f(x) = (13x^2 + 6x)/(12x^2 - 5)

Learn how to find the horizontal asymptote of the rational function f(x) = (13x^2 + 6x)/(12x^2 - 5). This step-by-step guide explains how to compare the degrees of polynomials and calculate the asymptote using the ratio of leading coefficients. Suitable for Grades 10-12.

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