One- And Two-Sided Limits, and When Limits Fail to
Exist
A function may not have a limit for a specific input value. The following
examples illustrate several cases.
Try the following:
- The first graph shows a jump discontinuity. What is the limit when c = 1? In other words, what value does f (x)
approach as x approaches 1? Move the x slider so that x gets closer and closer to 1. As you will note, f (x) approaches 1 as x approaches 1 from the left, but f (x) approaches 2 as x approaches 1 from the right.
We define a left-hand limit, written as
, to describe what happens as
we approach from the negative (left) direction. We define a right-hand
limit, written as 
, to describe what happens as we approach from the
positive (right) direction. If the left-hand and right-hand limits are
different (as in this case), then we say that the limit of f (x) at c does not exist. If the left-hand and
right-hand limits exist and are the same, then we say that the limit does
exist.
- Select the second example. This function has a vertical asymptote. What
is the limit when c = 1? In other words, what value does f (x) approach as x approaches 1? Move the x slider so that x gets closer and closer to 1. As you will note, f (x) approaches infinity from either direction. We say
that the limit is unbounded, or does not exist in this case, because
infinity is not a number. We sometimes write
to indicate that the limit
does not exist because it is unbounded.
- Select the third example. This function, sin(1/x), is
very wiggly around the origin. As you move the c slider around x = 0 you will notice that the y value jumps around. Zoom in
and try again. The more you zoom in, the more wiggly it looks (it isn't
really getting more wiggly, but rather that the graphing software has
limited resolution and is just showing more of the closely-spaced wiggles
as you zoom in). This function does not have a limit at c = 0
because it doesn't approach any single value.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.