Estimation and Limits -- Limits as x --> Infinity

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

We are often interested in the behavior of a function f(t) when t is very large -- either very large and positive or very large and negative. For example, if f(t) describes some quantity at time t then when t is very large and positive f(t) describes what happens in the far future. When t is very large and negative then f(t) describes what happens in the long ago past.

We use the notation

  Lim     f(t)
t --> +oo

to describe the behavior of f(t) for large positive values of t. and the notation

  Lim     f(t)
t --> -oo

to describe the behavior of f(t) for large negative values of t.

Consider, for example, the function

Missing equation

We can examine the behavior of this function for large positive and for large negative values of t three ways.

Numerically.
Graphically.
Algebraically.

For the first two ways your CAS window will be very useful.

Numerically: We can learn a lot about the behavior of a function for large positive values of t by computing its values for some large positive values of t -- for example, for this function we see that
```
f(10) = 0.017392

                     -18
f(100) = 2.45965 * 10
```
and based on this evidence it looks as if
```
  Lim     f(t) = 0
t --> +oo
```
We can learn a lot about the behavior of a function for large negative values of t by computing its values for some large negative values of t -- for example, for this function we see that
```
f(-10) = 3.00093
f(-100) = 3.0
```
and based on this evidence it looks as if
```
  Lim     f(t) = 3
t --> -oo
```
Graphically: The figure below shows this same function graphically.

Notice that for large positive values of t -- that is, at the extreme right of the graph -- f(t) is close to zero and for large negative values of t -- that is, at the extreme left of the graph -- f(t) is close to 3. This is exactly what we saw numerically above.
Algebraically: If we do a little algebra we can gain some additional insight into the behavior of the function f(t) for large positive or for large negative values of t.
First, notice that if t is large and negative then 2^t and 3^t are both very small. For example,
```
 -10
2    = 1/2^10 = 1/1024


 -10
3   = 1/3^10 = 1/59,049
```
Thus, the numerator of f(t) is close to 3 and the denominator of f(t) is close to 1. As a result f(t) is close to 3/1, which agrees with our earlier numerical and graphical observations.
To see what happens when t is large and positive we divide both the numerator and the denominator by 3^t to get

Now, we can see that when t is large and positive
```
 1 - t
3

     t
(2/3)
```
and
```
     t
(1/3)
```
are all close to zero, so the numerator of f(t) is close to 0 and its denominator is close to 1. The net result is that for large positive values of t, f(t) is close to 0 in agreement with our earlier numerical and graphical observations.

While numerical and graphical methods are important they can be misleading if they are used carelessly. Consider, for example, the function

Missing equation

Use your CAS window to graph this function, first using the range -10 <= t <= 10 for t and then using the range -1000 <= t <= 1000 for t .

Notice that using the first range it looks as if

  Lim     f(t) = 0 
t --> +oo

and

  Lim     f(t) = 3 
t --> -oo

but that with a larger range for t we see that for large positive and for large negative values of t, f(t) is large and positive. We write this

  Lim     f(t) = +oo 
t --> +oo

and

  Lim     f(t) = +oo 
t --> -oo

The reason for the misleading nature of the first graph is the term t^2/100,000. This term causes the function to become very large when t is either very large and positive or very large and negative but because of the denominator -- 100,000 -- the effect of this term is not felt unless t is quite large.

Check Your Understanding

For each of the following functions f(t) find

  Lim     f(t)
t --> +oo

and

  Lim     f(t)
t --> -oo

Note that the limit may be a real number or it may be +oo or -oo.