Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(4x²+25)/(x²+9)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{4x^{2} + 25}{x^{2} + 9}\).
2. Determine vertical asymptotes by finding values of \(x\) that make the denominator zero. Solve the equation \(x^{2} + 9 = 0\) for \(x\).
3. Since \(x^{2} + 9 = 0\) has no real solutions (because \(x^{2} = -9\) is not possible for real \(x\)), conclude that there are no vertical asymptotes.
4. To find horizontal or oblique asymptotes, compare the degrees of the numerator and denominator polynomials. Both numerator and denominator are degree 2.
5. Because the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients: \(y = \frac{4}{1} = 4\). There is no oblique asymptote since the degrees are equal.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero and the numerator is nonzero, causing the function to approach infinity or negative infinity. To find them, set the denominator equal to zero and solve for x. These lines indicate values that the function cannot take.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, compare the degrees of the numerator and denominator: if degrees are equal, the asymptote is the ratio of leading coefficients; if numerator degree is less, the asymptote is y=0; if greater, no horizontal asymptote exists.

Oblique (Slant) Asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They are found by performing polynomial long division of numerator by denominator. The quotient (without the remainder) gives the equation of the slant asymptote, representing the end behavior of the function.