7.6 Horizontal Asymptotes

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as the input (x) approaches positive or negative infinity.

Front

How do you determine the horizontal asymptote of a rational function?

Back

For a rational function of the form \( \frac{P(x)}{Q(x)} \), compare the degrees of the polynomials P and Q. If the degree of P is less than Q, the horizontal asymptote is \( y = 0 \). If the degree of P equals Q, the asymptote is \( y = \frac{a}{b} \) where a and b are the leading coefficients of P and Q, respectively. If the degree of P is greater than Q, there is no horizontal asymptote.

Front

What is the horizontal asymptote of the function \( f(x) = \frac{2x^2 + 3}{4x^2 + 5} \)?

Back

The horizontal asymptote is \( y = \frac{2}{4} = \frac{1}{2} \).

Front

What is the horizontal asymptote of the function \( f(x) = \frac{3x^3 + 2}{x^2 + 1} \)?

Back

There is no horizontal asymptote because the degree of the numerator (3) is greater than the degree of the denominator (2).

Front

If a function has a horizontal asymptote at \( y = 5 \), what does this imply about the function's behavior as \( x \) approaches infinity?

Back

It implies that as \( x \) approaches infinity, the function's values approach 5.

Front

What is the horizontal asymptote of the function \( f(x) = \frac{5}{x} \)?

Back

The horizontal asymptote is \( y = 0 \).

Front

For the function \( f(x) = \frac{x^2 - 1}{2x^2 + 3} \), what is the horizontal asymptote?

Back

The horizontal asymptote is \( y = \frac{1}{2} \).

Front

What does it mean if a function has no horizontal asymptote?

Back

It means that as \( x \) approaches positive or negative infinity, the function does not approach a specific finite value.

Front

Find the horizontal asymptote of the function \( f(x) = \frac{2x^2 + 3x + 1}{x^2 - 4} \).

Back

The horizontal asymptote is \( y = 2 \).

Front

What is the horizontal asymptote of the function \( f(x) = \frac{1}{x^2} \)?

Back

The horizontal asymptote is \( y = 0 \).