Calculus Lab Report

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Calculus: An Introduction to Limits

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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, 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
We will now present an amazingly useful result, even though we cannot prove it yet. We can find the limit at of any polynomial or rational function, as long as that rational function is defined at (so we are not dividing by zero). That is, must be in the domain of the function.
Limits of Polynomials and Rational functions
If is a polynomial or rational function that is defined at then

We already learned this for trigonometric functions, so we see that it is easy to find limits of polynomial, rational or trigonometric functions wherever they are defined. In fact, this is true even for combinations of these functions; thus, for example, .
The Squeeze Theorem

Graph showing being squeezed between and
The Squeeze Theorem is very important in calculus, where it is typically used to find the limit of a function by comparison with two other functions whose limits are known.
It is called the Squeeze Theorem because it refers to a function whose values are squeezed between the values of two other functions and , both of which have the same limit . If the value of is trapped between the values of the two functions and , the values of must also approach