Calculus: Introduction to Limits

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Calculus: An Introduction to Limits

Intuitive Look
A limit looks at what happens to a function when the input approaches a certain value. The general notation for a limit is as follows:

This is read as "The limit of  of  as  approaches ". We'll take up later the question of how we can determine whether a limit exists for  at  and, if so, what it is. For now, we'll look at it from an intuitive standpoint. Let's say that the function that we're interested in is , and that we're interested in its limit as  approaches . Using the above notation, we can write the limit that we're interested in as follows:

One way to try to evaluate what this limit is would be to choose values near 2, compute  for each, and see what happens as they get closer to 2. This is implemented as follows: | 1.7| 1.8| 1.9| 1.95| 1.99| 1.999|

| 2.89| 3.24| 3.61| 3.8025| 3.9601| 3.996001|
Here we chose numbers smaller than 2, and approached 2 from below. We can also choose numbers larger than 2, and approach 2 from above: | 2.3| 2.2| 2.1| 2.05| 2.01| 2.001|
| 5.29| 4.84| 4.41| 4.2025| 4.0401| 4.004001|
We can see from the tables that as  grows closer and closer to 2,  seems to get closer and closer to 4, regardless of whether  approaches 2 from above or from below. For this reason, we feel reasonably confident that the limit of  as  approaches 2 is 4, or, written in limit notation,

We could have also just substituted 2 into  and evaluated: . However, this will not work with all limits. Now let's look at another example. Suppose we're interested in the behavior of the function  as  approaches 2. Here's the limit in limit notation:

Just as before, we can compute function values as  approaches 2 from below and from above. Here's a table, approaching from below: | 1.7| 1.8| 1.9| 1.95| 1.99| 1.999|
| -3.333| -5| -10| -20| -100| -1000|
And here from above:
| 2.3| 2.2| 2.1| 2.05| 2.01| 2.001|
| 3.333| 5| 10| 20| 100| 1000|
In this case, the function doesn't seem to be approaching a single value as  approaches 2, but instead becomes an extremely large positive or negative number (depending on the direction of approach). This is known as an infinite limit. Note that we cannot just substitute 2 into  and evaluate as we could with the first example, since we would be dividing by 0. Both of these examples may seem trivial, but consider the following function:

This function is the same as

Note that these functions are really completely identical; not just "almost the same," but actually, in terms of the definition of a function, completely the same; they give exactly the same output for every input. In algebra, we would simply say that we can cancel the term , and then we have the function . This, however, would be a bit dishonest; the function that we have now is not really the same as the one we started with, because it is defined when , and our original function was specifically not defined when . In algebra we were willing to ignore this difficulty because we had no better way of dealing with this type of function. Now, however, in calculus, we can introduce a better, more correct way of looking at this type of function. What we want is to be able to say that, although the function doesn't exist when , it works almost as though it does. It may not get there, but it gets really, really close. That is, . The only question that we have is: what do we mean by "close"? -------------------------------------------------

Informal Definition of a Limit
As the precise definition of a limit is a bit technical, it is easier to start with an informal definition; we'll explain the formal definition later. We suppose that a function  is defined for  near  (but we do not require that it be defined when ). Definition: (Informal definition of a limit)

We call  the limit...
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