Mat210 Section 1.1  A Dash of Limits

Why study limits?

We need to have a basic understanding of "closeness" before we can dive into calculus. This type of concept is the study of limits.

What is a limit?

We deal with limiting values all of the time:

just to name a few examples. Intuitively a limit in every day life is a value we can get very close to, maybe even reach, but we ought not to exceed. The mathematical definition of a limit goes further than that. There are very specific constraints we put on the idea of a limit to be able to determine whether a limit exists or not. We go back to the input-ouput idea of functions and ask, for a given set of inputs x, approaching a given value "a", for x values both greater than and less than a, do the outputs y approach a certain value "A"? If so, the output value y = A is the number that we call the limit.

Let's consider the notation for the limit of f(x) as x gets sufficiently close to (but not equal to) a, where the answer to this limit would be equal to A.

There are three ways we can look at limits: algebraically, graphically and numerically. When considering a problem algebraically, we can apply the Rules for Limits. These methods are illustrated in the following examples.

Example 1: Find the limit numerically:  by creating a table of values for f(x) as x gets closer and closer to zero.


 

x

1

0.1

0.01

.001

       0

  -0.001

-0.01

-0.1

y=f(x)

undefined

After calculating the y values in the table, roll your mouse over the images to check your answers. You should observe the number pattern for the y values: they approach the same number as x gets close to zero. IF THE NUMBER PATTERN FOR THE Y VALUES DOES NOT APPROACH THE SAME VALUE FROM BOTH SIDES, THEN THE LIMIT DOES NOT EXIST. We will learn more about this in section 1.2, but keep this point in mind for today's homework (hint hint!)

Thus we conclude that  = even though the function itself is undefined at x = 0 !!! 

Rules for Limits:

There are a few examples where it seems as if we can obtain an answer by simply plugging the value x is approaching right into the function. That really is the case when we use the rules for limits. These are listed in the text on page 3. We can use rules (a) and (b) for the next problem. Eventually, you might just pop that value in for x and not actually use the rules. As long as you are aware of what's going on, this is an acceptable way to solve the limit.

Example 2: Find
Here we can use the Rules for Limits:
So the limit is equal to 0.

Example 3: Find


 

Factor, cancel, and evaluate: 

Verify Graphically

We can not see the “hole” at x =4, but our calculator verifies that it is there.

Verify Numerically

We can see that as x gets close to 4 from either direction, y gets close to 7.

So the answer is, the limit of the function as x approaches 4 is 7.

Example 4: Find 


 

Same for any (real number) constant value, 

Verify Graphically

Verify Numerically


  So in general, the limit of a constant is always that constant!

 

Last Update: January 9, 2010 la copyright 2010 (c) Leslie Arce and theDepartment of Mathematics and Statistics at ASU