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

How do you find vertical asymptotes of a rational function?

Back

To find vertical asymptotes of a rational function, set the denominator 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 does it mean if the left-hand limit and right-hand limit at a vertical asymptote are both infinite?

Back

If both the left-hand limit and right-hand limit at a vertical asymptote are infinite, it indicates that the function approaches positive or negative infinity from both sides of the asymptote.

Front

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

Back

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

Front

Define left-hand limit.

Back

The left-hand limit of a function at a point is the value that the function approaches as the input approaches that point from the left side.

Front

Define right-hand limit.

Back

The right-hand limit of a function at a point is the value that the function approaches as the input approaches that point from the right side.

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, typically due to division by zero.

Front

Provide an example of a function with a vertical asymptote.

Back

The function f(x) = 1/(x-2) 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 the asymptote line but will never touch or cross it, often diverging to positive or negative infinity.

Front

How can you determine the behavior of a function near a vertical asymptote?

Back

To determine the behavior of a function near a vertical asymptote, analyze the left-hand and right-hand limits as the input approaches the asymptote.