Vertical Asymptotes and Holes in Functions

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 that the function is undefined 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 they do not also make the numerator zero.

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 undefined 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-value of the hole.

Front

What is the significance of a hole in a graph?

Back

A hole in a graph indicates a point where the function is not defined, but the function approaches a specific value as it nears that point.

Front

If a function has a hole at x = a, what does this imply about the function's limit as x approaches a?

Back

The limit of the function as x approaches a exists and is equal to the value of the function at that point, except the function is not defined at x = a.

Front

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

Back

Vertical asymptotes indicate that the function will increase or decrease without bound as it approaches the asymptote, leading to infinite values.

Front

How many vertical asymptotes can a rational function have?

Back

A rational function can have multiple vertical asymptotes, depending on the degree of the polynomial in the denominator.

Front

What is the difference between a vertical asymptote and a hole?

Back

A vertical asymptote indicates a point where the function approaches infinity, while a hole indicates a point where the function is undefined but approaches a finite limit.

Front

Back

The hole is at x = -5, as it is a common factor in both the numerator and denominator.