(1)
Many mathematical models for physical phenomena involve smoothly varying quantities. For example, you may have seen that a body dropped from rest at a height h at time t = 0 will be at position
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at time t. (g is the gravitational constant - approximately 9.8 kg m/sec2) For simplicity, let's assume that m g = 1 and h = 2, so (1) simplifies to
How fast is the object moving at time t = 1?
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You can see from the graph that we are not dealing with an object moving at a constant speed. If it were moving at a constant speed, the plot would be a straight line. Instead, the object accelerates as at it falls, moving slowly at first and more quickly as it approaches the ground. Fortunately, the acceleration is not so great that the velocity changes greatly over short intervals of time. If we focus our attention near the time t = 1, we can see that plot looks almost like a straight line.
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Given that the plot looks almost like a straight line, we can estimate the velocity near t = 1 by assuming for the moment that the object is moving with a constant velocity. This greatly simplifies things. To compute the speed of an object moving with a constant speed we can compute
To estimate the velocity at t = 1, we could compute
If the object were moving with constant speed this would be the end of the story. However, the point is that this object is not moving at a constant speed. The result of -1 is at best an approximation. We can see that the issue is not settled by doing another estimate using slightly different points in time.
This is close to -1, but is not exactly equal to -1. To resolve the issue to our satisfaction we need a more sophisticated way to calculate the velocity. In order to make the calculation more general, we have to get more abstract. What we want is an estimate for the velocity over some brief interval of time [t1 , t2].
Substituting the expression for x(t) we get
which simplifies to
and subsequently to
What happens when t1 and t2 get very close to 1? You can see immediately from the final form for the velocity estimate that as the two points in time approach 1 the velocity approaches -1. Thus we say that in the limit of a very small time interval around t = 1 the velocity is -1.
The numerical technique we used to estimate the limit in the example is a useful general method for computing limits. If you want to know the value of some function as the independent variable approaches some limiting value, you can compute the function at a series of points that fall ever closer to the limit point. In the vast majority of cases, you can spot a trend in the values and make an educated guess concerning the limit.
Here are some simple examples. First, consider the polynomial function
To compute the limiting value of y as x approaches, say, -1, we can try the obvious option - go right to x = -1 and compute y. When we do that we see that y is 1. For the purpose of illustration, let us assume for the moment that we could not do the obvious thing and had to approach the problem indirectly. We would then select some points that fall ever closer to -1 and look for a trend.
| x | y |
|---|---|
| -1.02 | 0.91717599999999999 |
| -1.01 | 0.95929700000000007 |
| -1.005 | 0.97982462500000045 |
| -1.003 | 0.98793691900000025 |
| -1.00002 | 0.99991999719997593 |
| -1.0000001 | 0.99999959999992982 |
This indicates that the limiting value is y = 1. This is not the end of the story, because we also have to confirm that as we approach -1 from the right we see a similar limit.
| x | y |
|---|---|
| -0.9 | 1.3329999999999997 |
| -0.95 | 1.1828750000000001 |
| -0.975 | 1.0956718750000001 |
| -0.99 | 1.0393030000000001 |
| -0.999 | 1.0039930030000004 |
| -0.9999 | 1.0003999300029998 |
Finally, we can get even more confirmation by making a plot of the function.
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A more compact way to write the result is the following
which we read “the limit of y as x approaches -1 is 1.”
Starting now we are going to use the limit notation introduced at the end of the last example. A limit expression has the following general form:
The limit expression asks about the value of the function (y-values) as the value of the independent variable x approaches some limiting value a. If we can say that as x gets ever closer to 
gets ever closer to L, then we say that f has a limit of L at a.
Not every example is as simple as the last one. Much more frequently in the days to come we are going to run across examples that look more like this one. The function in this example is
Again the problem is to determine what happens as x approaches a limiting value. What happens near x = 0? This time around the obvious option of simply plugging in x = 0 is not available to us. At x = 0, the function has value 0/0, which is indeterminate. We can still estimate the limit by computing the value of the function at a sequence of points that fall ever closer to 0.
| x | y |
|---|---|
| 0.2 | -1.5003287933441996 |
| 0.1 | -1.7095923609751362 |
| 0.05 | -1.8420123198547096 |
| 0.003 | -1.9896673541629299 |
| -0.002 | -2.0069549719003446 |
| -0.00001 | -2.0000346416828307 |
This indicates that the limiting value is -2. A plot confirms this.
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As we have seen in the examples above, there are three techniques that are typically used to compute limits
in cases where the function f(x) may not be defined at x = a. The three techniques are
Each of these methods has advantages and disadvantages. Let's look at each on in turn.
This method works well provided you have the appropriate computing resources, such as a good calculator or computer software. For example, to compute the limit
we could use some software to make a table:
| x |  |
|---|---|
| -0.1 | .9983341664682815 |
| -0.01 | .9999833334166664 |
| -0.0005 | .9999999583333337 |
| 0.00001 | .9999999999998333 |
| 0.0002 | .9999999933333334 |
| 0.01 | .9999833334166664 |
| 0.05 | .9995833854135665 |
| 0.1 | .9983341664682815 |
This seems like reasonable evidence to support the claim that the limit is 1.
It is possible to be fooled by this method, especially if the problem exposes limitations in your calculator or software. For example, consider
Here is a similar table for this function near 0.
| x |  |
|---|---|
| -0.1 | .1666203960726697 |
| -0.01 | .1666662037047572 |
| -0.0005 | .1666666644695169 |
| 0.00001 | .1666666804567284 |
| 0.0000000001 | 0 |
| 0.0002 | .1666666693544982 |
| 0.01 | .1666662037047572 |
| 0.05 | .1666550941997968 |
| 0.1 | .1666203960726697 |
There is pretty strong evidence here that the limit is 0.1666
, but one of the entries looks odd. What happened in the case of x = 0.0000000001 was that computing x2 + 9 created a number that mathematically should be 9.00000000000000000001. Most calculators and computer programs can not handle that many digits of accuracy and simply round off this number to 9. That explains why the result comes out 0.
If you have a graphing calculator or good graphing software, you can use that to gather evidence for a limit having a particular value. If we plot the last example near 0 we see something like this:
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The problem here can lie in interpreting the results of the plot. It is somewhat difficult to read off the result that this limit is 1/6. You may be able to get better result by zooming in on the point of interest.
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The one thing you have to watch out for here is that if you zoom in too close the same numerical problems that arose in the last example will pop up again.
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By far the best method for computing a difficult limit is to use algebra to simplify the problem. The last example we looked at is well suited to this technique (provided you are sufficiently clever to see what to do):
The latter limit is easy to do:
This establishes conclusively that the limit is 1/6.
The problem with the algebraic simplification method (beyond requiring cleverness) is that some problems simply do not lend themselves to algebraic simplification. Here are two examples:
All of the examples so far may lead you to believe that in every case where a function as a 0/0 problem at some point the function has a well-defined, finite limit as you approach that point. Unfortunately, that is not true, as the example below demonstrates:
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As x approaches -1 from either side the function becomes extremely large. To further complicate things, the function becomes infinitely large in the negative direction as we approach from the left and infinitely large in the positive direction as we approach from the right. We can summarize this information by giving a pair of one-sided limits.
The first limit is read "The limit as x approaches -1 from below of 
is negative infinity." The second limit is read "The limit as x approaches -1 from above of 
is positive infinity."
Section 2.1: 3, 4
Section 2.2: 4, 5, 6, 17, 18, 26, 27