Limits That Do Not Exist

    This topic illustrates when a limit does not exist by giving three case examples:

    (1) A limit does not exist because the one-sided limits do not agree in value.
    
    (2) A limit does not exist because of an oscillating behavior of a function, and
    
    (3) a limit does not exist (no finite value) because of an unbounded behavior of a function.

    Consider, for example, a piecewise function with a jump where the function is pieced together (defined or not). Even though the one-sided limits might exist, they must agree in value for the two-sided limit to exist. In short, if the one-sided limits do not agree then the two-sided limit does not exist.
    Also, consider another case where a function has an oscillating behavior. On one hand, the trigonometric functions all have an oscillating (periodic) behavior. However, imagine a function where the oscillation becomes much more pronounced as the variable approaches a fixed point; this type of oscillating behavior is where the function may not have a limit.
    Finally, we illustrate the case where a function becomes unbounded as the variable approaches a fixed point; for example, a function with a vertical asymptote. Without a finite number to assign the limit, we sometimes say that the limit does not exist.

Example (Limit Does Not Exist Left and Right Behavior) Find for

.

    Solution. The limit does not exist because



which can be seen from the graph:

Example (Limit Does Not Exist Oscillating Behavior) Find


    Solution. The limit does not exist because   and   for

  and   

respectively. The graph of   is oscillating around so we infer that the limit does not exist because does not approach a number, but rather oscillates, as approaches 0. This behavior can be seen in a good sketch of the graph:

Example (Infinite Limits) Determine

    Solution. Since decreases without bound as and   increases without bound as we say that does not exist.

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Cite this as:
Limits That Do Not Exist
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/limits-that-do-not-exist.html