Questions: Find the horizontal and vertical asymptotes. f(x)=(7x^2+36)/(x^2-36)

Solution Steps

To find the horizontal and vertical asymptotes of the function \( f(x) = \frac{7x^2 + 36}{x^2 - 36} \):

1. Vertical Asymptotes: These occur where the denominator is zero and the numerator is not zero. Solve \( x^2 - 36 = 0 \) to find the vertical asymptotes.
2. Horizontal Asymptotes: These are determined by the degrees of the polynomials in the numerator and the denominator. Since the degrees of the numerator and the denominator are the same (both are 2), the horizontal asymptote is the ratio of the leading coefficients.

To find the horizontal asymptote of the function \( f(x) = \frac{7x^2 + 36}{x^2 - 36} \), we compare the degrees of the numerator and the denominator.

• The degree of the numerator \( 7x^2 + 36 \) is 2.
• The degree of the denominator \( x^2 - 36 \) is also 2.

Since the degrees of the numerator and the denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients.

\[ \text{Horizontal Asymptote: } y = \frac{7}{1} = 7 \]

To find the vertical asymptotes, we set the denominator equal to zero and solve for \( x \).

\[ x^2 - 36 = 0 \]

Solving for \( x \):

\[ x^2 = 36 \]

\[ x = \pm 6 \]

Thus, the vertical asymptotes are at \( x = 6 \) and \( x = -6 \).

Final Answer