In the previous section we looked at a couple of problems and in both problems we had a function (slope in the tangent problem case and average rate of change in the rate of change problem) and we wanted to know how that function was behaving at some point .  At this stage of the game we no longer care where the functions came from and we no longer care if we’re going to see them down the road again or not.  All that we need to know or worry about is that we’ve got these functions and we want to know something about them.

To answer the questions in the last section we choose values of x that got closer and closer to  and we plugged these into the function.  We also made sure that we looked at values of x that were on both the left and the right of .  Once we did this we looked at our table of function values and saw what the function values were approaching as x got closer and closer to  and used this to guess the value that we were after.

This process is called taking a limit and we have some notation for this.  The limit notation for the two problems from the last section is,

In this notation we will note that we always give the function that we’re working with and we also give the value of x (or t) that we are moving in towards.

In this section we are going to take an intuitive approach to limits and try to get a feel for what they are and what they can tell us about a function.  With that goal in mind we are not going to get into how we actually compute limits yet.  We will instead rely on what we did in the previous section as well as another approach to guess the value of the limits.

Both of the approaches that we are going to use in this section are designed to help us understand just what limits are.  In general we don’t typically use the methods in this section to compute limits and in many cases can be very difficult to use to even estimate the value of a limit and/or will give the wrong value on occasion.  We will look at actually computing limits in a couple of sections.

Let’s first start off with the following “definition” of a limit.

Definition

This is not the exact, precise definition of a limit.  If you would like to see the more precise and mathematical definition of a limit you should check out the The Definition of a Limit section at the end of this chapter.  The definition given above is more of a “working” definition.  This definition helps us to get an idea of just what limits are and what they can tell us about functions.

So just what does this definition mean?  Well let’s suppose that we know that the limit does in fact exist.  According to our “working” definition we can then decide how close to L that we’d like to make f(x).  For sake of argument let’s suppose that we want to make f(x) no more that 0.001 away from L.  This means that we want one of the following

Now according to the “working” definition this means that if we get x sufficiently close to we can make one of the above true.  However, it actually says a little more.  It actually says that somewhere out there in the world is a value of x, say X, so that for all x’s that are closer to a than X then one of the above statements will be true. 

This is actually a fairly important idea.  There are many functions out there in the work that we can make as close to L for specific values of x that are close to a, but there will other values of x closer to a that give functions values that are nowhere near close to L.  In order for a limit to exist once we get f(x) as close to L as we want for some x then it will need to stay in that close to L (or get closer) for all values of x that are closer to a.  We’ll see an example of this later in this section.

In somewhat simpler terms the definition says that as x gets closer and closer to x=a (from both sides of course…) then f(x) must be getting closer and closer to L.  Or, as we move in towards x=a then f(x) must be moving in towards L.

It is important to note once again that we must look at values of x that are on both sides of x=a.  We should also note that we are not allowed to use x=a in the definition.  We will often use the information that limits give us to get some information about what is going on right at x=a, but the limit itself is not concerned with what is actually going on at x=a.  The limit is only concerned with what is going on around the point x=a.  This is an important concept about limits that we need to keep in mind.

An alternative notation that we will occasionally use in denoting limits is

How do we use this definition to help us estimate limits?  We do exactly what we did in the previous section.  We take x’s on both sides of x=a that move in closer and closer to a and we plug these into our function.  We then look to see if we can determine what number the function values are moving in towards and use this as our estimate.

Let’s work an example.

Let’s think a little bit more about what’s going on here.  Let’s graph the function from the last example.  The graph of the function in the range of x’s that were interested in is shown below.

First, notice that there is a rather large open dot at .  This is there to remind us that the function (and hence the graph) doesn’t exist at .

As we were plugging in values of x into the function we are in effect moving along the graph in towards the point as .  This is shown in the graph by the two arrows on the graph that are moving in towards the point.

When we are computing limits the question that we are really asking is what y value is our graph approaching as we move in towards  on our graph.  We are NOT asking what y value the graph takes at the point in question.  In other words, we are asking what the graph is doing around the point .  In our case we can see that as x moves in towards 2 (from both sides) the function is approaching  even though the function itself doesn’t even exist at .  Therefore we can say that the limit is in fact 4.

So what have we learned about limits?  Limits are asking what the function is doing around  and are not concerned with what the function is actually doing at .  This is a good thing as many of the functions that we’ll be looking at won’t even exist at  as we saw in our last example.

Let’s work another example to drive this point home.

Let’s make the point one more time just to make sure we’ve got it.  Limits are not concerned with what is going on at . Limits are only concerned with what is going on around .  We keep saying this, but it is a very important concept about limits that we must always keep in mind.  So, we will take every opportunity to remind ourselves of this idea.

Since limits aren’t concerned with what is actually happening at  we will, on occasion, see situations like the previous example where the limit at a point and the function value at a point are different.  This won’t always happen of course.  There are times where the function value and the limit at a point are the same and we will eventually see some examples of those.  It is important however, to not get excited about things when the function and the limit do not take the same value at a point.  It happens sometimes and so we will need to be able to deal with those cases when they arise.

Let’s take a look another example to try and beat this idea into the ground.