Find the horizontal asymptote of each function. f(x)=x2+4x2x3+8x2f\left(x\right)=\frac{x^2+4x}{2x^3+8x^2}f(x)=2x3+8x2x2+4x

Horizontal Asymptote at y = 0 y=0 y = 0

Find the horizontal asymptote of each function. f ( x ) = x 2 + 4 x 2 x 3 + 8 x 2 f\left(x\right)=\frac{x^2+4x}{2x^3+8x^2} f ( x ) = 2 x 3 + 8 x 2 x 2 + 4 x

In this problem, we're asked to find the horizontal asymptote of the function f of x is equal to x squared plus 4x over 2x cubed plus 8x squared. So here, looking at the degree of my numerator and denominator, the degree of my numerator is 2. The degree of my denominator is 3. So that tells me that the degree of my numerator is less than the degree of my denominator, which means my work is done. I simply have a horizontal asymptote at y equals 0.

Find the horizontal asymptote of each function. f ( x ) = x 2 + 4 x 2 x 3 + 8 x 2 f\left(x\right)=\frac{x^2+4x}{2x^3+8x^2} f ( x ) = 2 x 3 + 8 x 2 x 2 + 4 x

Horizontal Asymptote at y = 0 y=0 y = 0

Horizontal Asymptote at y = 1 2 y=\frac12 y = 2 1

Horizontal Asymptote at y = 2 y=2 y = 2

5. Rational Functions

Asymptotes

College Algebra

5. Rational Functions

Asymptotes

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Multiple Choice

Find the horizontal asymptote of each function.
$f\left(x\right)=\frac{x^2+4x}{2x^3+8x^2}$

Horizontal Asymptote at $y=0$

Horizontal Asymptote at $y=\frac12$

Horizontal Asymptote at $y=2$

Verified step by step guidance

Identify the degrees of the numerator and the denominator. The degree of the numerator is the highest power of x, which is 2 (from x^2), and the degree of the denominator is 3 (from 2x^3).

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 degrees were equal, the horizontal asymptote would be the ratio of the leading coefficients. However, since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

Consider the possibility of other horizontal asymptotes by analyzing the behavior of the function as x approaches infinity. Since the degree of the numerator is less than the degree of the denominator, the function approaches zero.

Conclude that the horizontal asymptote for the given function f(x) = \frac{x^2 + 4x}{2x^3 + 8x^2} is y = 0.

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