Infinite Limits and Vertical Asymptotes

    Infinite limits are used to described unbounded behavior of a function near a given real number which is not in the domain of the function. They are particularly useful for showing the intentions of the graph of a function by drawing dashed lines representing unbounded growth which are called vertical asymptotes.
    First we will define infinite limits and then state a proposition which will be illustrated through some examples. Then we define vertical asymptotes and give examples of functions that have vertical asymptotes, looks like it has a vertical asymptote but doesn't, and of a function that has infinitely many vertical asymptotes.

Definition (Infinite Limit) Let be a function defined on both sides of except possible at it self. Then



means that the values of can be made arbitrarily large by taking sufficiently close to   ( ).

Definition (Infinite Negative Limit) Let be a function defined on both sides of except possible at itself. Then



means that the values of can be made arbitrarily large negative by taking sufficiently close to ( ).

Proposition (Infinite Limit Theorem) Let be a positive real number.

    (i) If is a positive even integer, then



    (ii) If is a positive odd integer, then

Example (Infinite Limit) Evaluate the limit,  

    Solution. Notice that increases without bound as and therefore,

Example (Infinite Limit) Evaluate the limit,  

    Solution. Note that and decreases without bound as and therefore, This is true because when is close to we know that is negative. Similarily,   because when and close to we know that is negative.

    Vertical asymptotes are especially important when sketching rational and trigonometric functions. But keep in mind only one of following needs to hold for a function to have a vertical asymptote.

Definition (Vertical Asymptote) The line is called a vertical asymptote of the curve if at least one of the following statements is true:

    The following example demonstrates that not all rational functions have vertical asymptotes; and in fact, there can be an unlimited number of vertical asymptotes for a function.

Example (Vertical Asymptote) Determine the vertical asymptotes of the functions,

    Solution. Since,



and therefore, the vertical asymptotes are and because,



Example (Vertical Asymptote) Determine the vertical asymptotes of the functions,

    Solution. The function is continuous on its domain which is and therefore, there are no vertical asymptotes for this function.

Example (Vertical Asymptote) Determine the vertical asymptotes of the functions,

    Solution. We can write this function has
    


Therefore, the zeros of the sine and cosine functions yield the vertical asymptotes of for all integers The graph is

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Cite this as:
Infinite Limits And Vertical Asymptotes
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/infinite-limits-and-vertical-asymptotes.html