We are tasked with finding the horizontal and vertical asymptotes of the function:

$f(x) = \frac{x^2 + 4x - 5}{3x^2 + 14x - 5}$

Vertical Asymptotes:

Vertical asymptotes occur when the denominator is equal to zero (and the numerator is non-zero at that point). So, we need to solve the equation:

$3x^2 + 14x - 5 = 0$

We can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = 3$, $b = 14$, and $c = -5$. Plugging in these values:

$x = \frac{-14 \pm \sqrt{14^2 - 4(3)(-5)}}{2(3)}$ $x = \frac{-14 \pm \sqrt{196 + 60}}{6}$ $x = \frac{-14 \pm \sqrt{256}}{6}$ $x = \frac{-14 \pm 16}{6}$

Thus, the two solutions are:

$x = \frac{-14 + 16}{6} = \frac{2}{6} = \frac{1}{3}, \quad x = \frac{-14 - 16}{6} = \frac{-30}{6} = -5$

So, the vertical asymptotes are at $x = \frac{1}{3}$ and $x = -5$.

Horizontal Asymptotes:

For rational functions, the horizontal asymptote is determined by comparing the degrees of the polynomials in 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 degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

In this case, both the numerator and the denominator are degree 2 polynomials. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 3. Therefore, the horizontal asymptote is:

$y = \frac{1}{3}$

Final Answer:

• Horizontal asymptote: $y = \frac{1}{3}$
• Vertical asymptotes: $x = \frac{1}{3}, x = -5$

Would you like further details or explanations? Here are 5 related questions:

1. What is the process for finding slant asymptotes if the degrees of the polynomials differ?
2. How do we determine if a rational function has any holes?
3. Can we graph this function based on its asymptotes?
4. How does long division help in finding asymptotes for improper rational functions?
5. What is the behavior of the function near its vertical asymptotes?

Tip: Always check for any common factors between the numerator and the denominator, as these might indicate removable discontinuities (holes) instead of vertical asymptotes.