Horizontal/Vertical Asymptotoes/Holes

Front

What is a vertical asymptote?

Back

A vertical asymptote is a line x = a where a function approaches infinity or negative infinity as x approaches a. It occurs when the denominator of a rational function equals zero and the numerator does not equal zero at that point.

Front

How do you find vertical asymptotes in a rational function?

Back

To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x. The values of x that make the denominator zero are the vertical asymptotes, provided the numerator is not also zero at those points.

Front

What is a hole in a function?

Back

A hole occurs in a function at a point where both the numerator and denominator are zero, indicating that the function is not defined at that point, but the limit exists.

Front

How do you identify holes in a rational function?

Back

To identify holes, factor both the numerator and denominator. If a common factor exists, set it equal to zero to find the x-coordinate of the hole.

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a line y = b where a function approaches the value b as x approaches infinity or negative infinity. It describes the end behavior of the function.

Front

How do you find horizontal asymptotes in rational functions?

Back

To find horizontal asymptotes, compare the degrees of the numerator and denominator: 1) If the degree of the numerator is less than the degree of the denominator, y = 0 is the horizontal asymptote. 2) If the degrees are equal, divide the leading coefficients. 3) If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What is the significance of the degree of the numerator and denominator in determining horizontal asymptotes?

Back

The degree of the numerator and denominator determines the end behavior of the function. It helps identify whether the function approaches a specific value (horizontal asymptote) or diverges (no horizontal asymptote).

Front

Back

The vertical asymptote is x = 3, as the denominator equals zero at this point.

Front

What happens to a function at a vertical asymptote?

Back

As the function approaches the vertical asymptote, the function values increase or decrease without bound, approaching positive or negative infinity.

Front

What is the relationship between vertical asymptotes and the denominator of a rational function?

Back

Vertical asymptotes are determined by the values of x that make the denominator equal to zero, provided the numerator does not also equal zero at those points.