One sides, two sided, and nonexistent limits
A function may not have a limit for a specific input value.
The following examples illustrate several cases.
1. A jump discontinuity
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 lefthand limit, written as
to describe what happens as
we approach from the negative (left) direction.
We define a righthand limit, written as
to describe what happens as we approach from the
positive (right) direction. If the lefthand and righthand limits are
different (as in this case), then we say that the limit of f (x) at c does not exist. If the lefthand and
righthand limits exist and are the same, then we say that the limit does
exist.
2. A vertical asymptote
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.
3. A wiggly function sin(1/x)
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 closelyspaced wiggles
as you zoom in). This function does not have a limit at c = 0
because it doesn't approach any single value.
While you are here..
... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone.
However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site?
When we reach the goal I will remove all advertising from the site.
It only takes a minute and any amount would be greatly appreciated.
Thank you for considering it! – John Page
Become a patron of the site at patreon.com/mathopenref
Other differentiation topics
Acknowledgements
Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.
