In the last section we saw how a two points on a curve can approach each other, and how a secant line can geometrically approach the tangent line at a point on the curve. In this section, we are looking at the calculation side of this concept of approaching something. In the table below, we examine the sine function for angles close to 30 degrees. Here we see numerically that the sine approaches one half. In the last lesson and this lesson as we consider how a function changes as we approach a value, we are really beginning to ge the concept of a mathematical limit.
 

Here we take a close look at the sine of an angle that gets closer and closer to 30 degrees. We see that the sine of the angles get closer and closer to .5.

Graphically, we can see this same concept of a limit. If we observe the graph and table of values for the function   we immediately see that . But it raises a curiosity about what happens close to zero (on either side of zero). The table and graph give a good representation of this behavior near zero.
 
 

We are purposely only looking at the 1st quadrant for this problem. As x gets closer and closer to zero, we see that y gets larger and larger, and in fact, at a larger and larger rate.  In limit notation, this is written: . This says that the limit of the f(x) as x approaches 0 from the right is infinity. (Which really means there is no limit).  As x gets larger and larger, y gets closer and closer to zero.  In limit notation, this is written: . This says that the limit of the f(x) as x approaches infinity is 0. Here we didn't have to say "from the left", because this is the default.  So this problem illustrates two different limits.

Looking at the graph, we see both limits even more clearly.

We have already seen one instance where there was no limit. In the function we spoke of above, , we found that there was no limit to how high the function would get. Below is another example of no limit, but this time the limit cannot be given the answer infinity.
 
 

	Consider the function shown at the left.
	Here is a graph of the function.
What is the following limit equal to?:	This is the problem to be solved.
	Here is a "closer" look at the function. Check the labels on the x-axis in comparison with those in the graph above.
	Here is yet a closer look. Again, look at the units of measure on the x-axis.
So, what is  equal to?	The function is forever oscillating between -1 and 1, and no matter how much we zoom in on the function, the oscillation happens more frequently. Here the limit does not exist.

	Given the function at the left. Find .
	As x approaches 1 from the left, it is clear that f(x) is approaching 2. So the "left-hand" limit is 2.
	As x approaches 1 from the rightt, it is clear that f(x) is approaching 2. So the "right-hand" limit is also 2.
But,	Clearly, f(x) is not continuous at x = 1.
Nevertheless,	Because from both sides, the limit as x approaches 1 is 2. Notice that a limit must not be the same as the value of the function of x as the function approaches x. Notice also that a function can have a limit at a point of discontinuity.

If you have the Journey Through Calculus CD, load and run MResources/Module 2/Basics of Limits/Sound of a Limit that Does Not Exist. This module will allow you to listen to the sound of a function trying to approach a limit.

Using your Tools for Enriching Calculus CD (that came with your book), load and run Module 2.2. This module will help you to explore limits at points where graphs exhibit unusual behavior.

Intuitive Definition of a Limit

What is a Limit? @ Calculus-Help.com

When does a Limit exist? @ Calculus-Help.com

Limits @ Platonic Realms

Check Concepts