Vertical Asymptotes and Limits

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 does not have a defined value at x = a.

Front

What does it mean for a limit to be undefined?

Back

A limit is undefined when the function does not approach a specific finite value as the input approaches a certain point, often due to a vertical asymptote.

Front

What are left-hand and right-hand limits?

Back

Left-hand limit refers to the value that a function approaches as the input approaches a certain point from the left (x -> a-). Right-hand limit refers to the value that a function approaches as the input approaches from the right (x -> a+).

Front

How do you determine the existence of a vertical asymptote?

Back

To determine the existence of a vertical asymptote, find values of x that make the denominator of a rational function zero while ensuring the numerator is not zero at those points.

Front

What is the significance of the directions of limits at a vertical asymptote?

Back

The directions of limits (left-hand and right-hand) at a vertical asymptote indicate the behavior of the function as it approaches the asymptote, helping to understand whether it goes to positive or negative infinity.

Front

Provide an example of a function with a vertical asymptote.

Back

An example is f(x) = 1/(x-2), which has a vertical asymptote at x = 2.

Front

What happens to the graph of a function at a vertical asymptote?

Back

At a vertical asymptote, the graph of the function will approach infinity or negative infinity, creating a break in the graph.

Front

Can a function have more than one vertical asymptote?

Back

Yes, a function can have multiple vertical asymptotes, typically occurring at different values of x where the denominator is zero.

Front

What is the relationship between vertical asymptotes and discontinuities?

Back

Vertical asymptotes indicate points of discontinuity in a function where the function does not have a defined value.

Front

How do you find limits at vertical asymptotes?

Back

To find limits at vertical asymptotes, evaluate the left-hand and right-hand limits as the input approaches the asymptote from both sides.