Formal Limits for Vertical Asymptotes
- Dr. Jack L. Jackson II
When we are considering if a function f has a left or right limit as x approaches a real constant c, we are considering how the graph approaches the vertical line x = c. One way a left, right, or full limit of a function as x approaches a real constant c could fail to exist is when the graph of the function has a vertical asymptote from one or both sides. For example, if the graph is getting higher and higher with no bound as the value of x approaches c from the right then there is a vertical asymptote from that side, and we say the limit approaches infinity. Note that this is not really saying that the function has infinity as the limit, but rather describes how and why the limit does not exist. Formally, we have the following definition in this case. Formal Definition of a Vertical Asymptote from the right. We say that a function f has a vertical asymptote from the right and if and only if for every M > 0 there exists a > 0 such that if c < x < x + then f(x) > M. In the App: Enter a function for f(x) in its input box. Choose a value of c by typing in its input box. Choose a value of M by moving its slider or typing a value in its input box. Choose a corresponding value of delta () by adjusting its slider or typing in its input box. If the limit from the right is approaching infinity then the y -values corresponding to x-values in the interval (c, x + ) will be above M, i.e. the graph will not go out the bottom of the blue box. There are four colored boxes for right limit approaches infinity, right limit approaches negative infinity, left limit approaches infinity, and left limit approaches negative infinity. Can you see how to interpret each of these graphically? Can you see how to adjust the definition above for each of these four cases?