Vertical and Horizontal Asymptotes

Front

What is a vertical asymptote?

Back

A vertical asymptote is a line x = a where a function approaches infinity or negative infinity as the input approaches a. It indicates values that the function cannot take.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a line y = b that a function approaches as x approaches infinity or negative infinity. It indicates the behavior of the function at extreme values.

Front

How do you find vertical asymptotes in a rational function?

Back

Vertical asymptotes can be found by setting the denominator of a rational function equal to zero and solving for x.

Front

How do you find horizontal asymptotes in a rational function?

Back

Horizontal asymptotes can be determined by comparing the degrees of the numerator and denominator of a rational function.

Front

What is the horizontal asymptote if the degree of the numerator is less than the degree of the denominator?

Back

The horizontal asymptote is y = 0.

Front

What is the horizontal asymptote if the degree of the numerator is equal to the degree of the denominator?

Back

The horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator).

Front

What is the horizontal asymptote if the degree of the numerator is greater than the degree of the denominator?

Back

There is no horizontal asymptote; the function may have an oblique asymptote instead.

Front

Given the function f(x) = (2x^2 + 3)/(x^2 - 1), what is the horizontal asymptote?

Back

y = 2, since the degrees of the numerator and denominator are equal.

Front

Given the function f(x) = 1/(x - 3), what is the vertical asymptote?

Back

x = 3, since the denominator equals zero at this point.

Front

What does it mean if a function has a vertical asymptote at x = -5?

Back

It means that as x approaches -5, the function's value increases or decreases without bound.