Calculus/Finite Limits

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Informal Finite LimitsEdit

Now, we will try to more carefully restate the ideas of the last chapter. We said then that the equation   meant that, when   gets close to 2,   gets close to 4. What exactly does this mean? How close is "close"? The first way we can approach the problem is to say that, at   , which is pretty close to 4.

Sometimes however, the function might do something completely different. For instance, suppose   , so   . Next, if you take a value even closer to 2,   , in this case you actually move further from 4. The reason for this is that substitution gives us 4.23 as x approaches 2.

The solution is to find out what happens arbitrarily close to the point. In particular, we want to say that, no matter how close we want the function to get to 4, if we make   close enough to 2 then it will get there. In this case, we will write

 

and say "The limit of   , as   approaches 2, equals 4" or "As   approaches 2,   approaches 4." In general:

Definition: (New definition of a limit)

We call   the limit of   as   approaches   if   becomes arbitrarily close to   whenever   is sufficiently close (and not equal) to   .

When this holds we write

 

or

 

One-Sided LimitsEdit

Sometimes, it is necessary to consider what happens when we approach an   value from one particular direction. To account for this, we have one-sided limits. In a left-handed limit,   approaches   from the left-hand side. Likewise, in a right-handed limit,   approaches   from the right-hand side.

For example, if we consider   , there is a problem because there is no way for   to approach 2 from the left hand side (the function is undefined here). But, if   approaches 2 only from the right-hand side, we want to say that   approaches 0.

Definition: (Informal definition of a one-sided limit)

We call   the limit of   as   approaches   from the right if   becomes arbitrarily close to   whenever   is sufficiently close to and greater than   .

When this holds we write

 

Similarly, we call   the limit of   as   approaches   from the left if   becomes arbitrarily close to   whenever   is sufficiently close to and less than   .

When this holds we write

 

In our example, the left-handed limit   does not exist.

The right-handed limit, however,   .

It is a fact that   exists if and only if   and   exist and are equal to each other. In this case,   will be equal to the same number.

In our example, one limit does not even exist. Thus   does not exist either.

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Finite Limits