Limits: A Formal View
We have talked about limits using "approaches" when discussing how to
think informally about limits. We can make this concept more precise and
formal as follows.
- First, let's define some names. We will let L be the limit, c be the input value at which we want to find the limit, and f (x) be the function.
- Next, let's be more formal about what it means to approach c.
One way is to think about an interval around c that extends in the
positive and negative directions by a certain amount, which we will call
δ (delta). We could write this as c - δ < x < c + δ, or we can shorten this up somewhat to |x - c| < δ (i.e., x is within δ of c).
Since limits are about approaching c, not what the value is at c,
we will add the condition that x ≠ c. Also, δ is greater
than 0. Using this idea, approaching c means to shrink the
interval, or to make δ smaller.
- Similarly, we can talk about an interval around L that is
ε (epsilon) in either direction by saying |f (x) - L| < ε (i.e., f (x) is within ε
of L). We can think about approaching L as shrinking
ε (but, as above, ε is greater than 0).
- Now we need to relate the two intervals somehow. Think about a specific
ε, which gives us an interval of output values around L.
If L is the limit as x approaches c, then we should
be able to find some interval around c that keeps all of the
output values for that interval within ε of L. We might
have to make δ pretty small to do this, but if we can find one,
then things are looking good for L to be the limit, because we
have ruled out that output values head off to infinity as we approach c, and also that the output values don't wiggle or jump around
more than ε away from L. But, there still might be some
wiggling or jumping of the output values that is smaller than
- Lastly, let's allow ourselves to pick any ε we want, no matter
how small (but greater than 0). If we can still find a δ that keeps
all the output values within ε of L, then L must be
the limit, as we've ruled out any room for wiggles or jumps.
We can put all of this together into the formal definition of a limit:
We define as the number L such that
for every ε > 0, there exists a δ > 0 such that if
|x - c| < δ and x ≠ c, then |f (x) - L| < ε.
Try the following:
- The first graph show the line used in a previous example. Is the limit L = 0.5 when c = 1? The green vertical stripe represents
the interval of input values that are within δ of c, and the
pink horizontal stripe represents the output values that are within
ε of L. To use the formal definition we ask whether, for a
given setting of ε, we can find a δ such that the part of
the function that is within the green stipe also stays within the pink
stripe. The yellow box is the intersection of the two stripes, so we want
to know if any of the graph of the function "sticks out" of the yellow
box into the green stripe. As initially drawn, this example does have
some of the graph of the line extending into the green box. That means
that our δ is not small enough. Use the delta slider at the bottom
of the applet to shrink the green stripe until the graph doesn't stray
into the green stripe (it is okay for it to stray into the pink stripe). This means we have satisfied the condition in the
formal definition for one particular ε.
To be the limit, we have to be able to pick any ε that we want
and still be able to find a δ that works. Use the epsilon slider to
make ε smaller, such that some of the graph spills out into the
green stripe again. Can you find a δ using the delta slider that
keeps the graph out of the green part of the vertical stripe? You can see
that no matter how small you make ε, you can always find a
δ that works. You can even zoom in on the point at the center of
the yellow rectangle if you want to try a really small ε by
clicking on the center of the yellow rectangle (which zooms in by a
factor of 2). Hence the limit L = 0.5 when c = 1.
- Select the second example. This is just like the first case, except
that one point has moved. Is the limit still L = 0.5 when c = 1? If you play the same game as above, you find that for any ε
you pick, you can shrink δ down enough to keep the graph from
straying into the green, except for that displaced point. But that point
is not a problem, since we don't care about the value of f (c) when we are finding a limit. Recall that the formal definition
requires δ > 0 and x ≠ c, so that this point
doesn't count. Hence the limit L = 0.5 when c = 1.
- Select the third example. This is like the previous two cases, but
there is now a point missing. Is the limit still L = 0.5 when c = 1? This is just like the previous example, except that the
displaced point is gone. For any ε, you can shrink δ down
enough to keep the graph from straying into the green. Hence the limit L = 0.5 when c = 1.
- Select the fourth example. Is the limit L = 1 when c = 0?
Shrink δ until the graph doesn't stray into the green. Zoom in and
shrink ε, then find a new δ that works. Can you keep going,
always finding a δ for every ε? Yes, hence the limit L = 1 when c = 0.
- Select the fifth example, a jump discontinuity. Is the limit L =
1.5 when c = 1? You can shrink up δ as the graph is
initially shown to keep the function out of the green. But, if you shrink
ε to be smaller than 0.5, you will have a problem; no δ
will be small enough to keep the function out of the green. Hence the
limit is not 1.5. Might it be some other value? Try setting L to 2
using the L slider or input box. As you can see, this doesn't work
either. If you move L around, you will see that nothing works. No
matter what value you pick to try as the limit, there is an ε
such that the graph always sticks out into the green, no matter what
δ you try. Hence there is no limit at c = 1.
- Select the sixth example, a function with a vertical asymptote. What is
the limit when c = 1? Its obvious that it can't be 1.5, because
you can shrink δ as much as you like and the graph will still stick
up into the green (although for small δ the part that sticks into
the green may be outside of the window, in which case you can pan the
graph using the right mouse button to see this). Changing L won't
help; no matter how big you make L, some part of the graph will
still stick up into the green. Hence there is no limit at c =
- Select the seventh example, the wiggly sin(1/x). Is the
limit L = 0 when c = 0? As initially shown with a very wide
ε of 1.5, any δ at all works, because the graph always
stays in the pink. But, if you shrink ε to be less than 1, the
graph wiggles up into the green. Note that because of the limited
resolution of the graph, you might think you can narrow δ enough,
but zoom in and you will see that, in fact, the graph keeps wiggling
enough to stick up and down into the green. Hence 0 is not the limit at c = 0. Does some other value work? If you play around with other
values for L, you will find that none of them work, either. There
is always some ε that's small enough to cause the wiggles to
stick into the green, no matter what δ you try. Hence there is no
limit at c = 0.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.