F. Infinite Limits

We next look at functions whose limit at x=a does not exist, but whose values increase or decrease without bound as x approaches a from the left or right.

Infinite Limit (Useable Definition).

In general, we write

if we can make the value of f(x) arbitrarily large by taking x to be sufficiently close to aa (on either side of aa) but not equal to a. Similarly, we write

if we can make the value of f(x) arbitrarily large and \blue{negative} by taking x to be sufficiently close to a (on either side of a) but not equal to a.

Note:

1. We want to emphasize that by the proper definition of limits, the above limits do not exist, since they are not real numbers. However, writing ±∞ provides us with more information than simply writing DNE.

2. This definition can be modified for one-sided limits as well as limits with x →a replaced by x→∞ or x→−∞.

Example: Infinite limit