Limits involving infinity.html

Limits Involving Infinity

Infinite Limits

Look at the following examples. In each case construct the graph of the function and a table of values.

Example 1

What is ?

Solution

Example 2

The graph of is shown. What are , , and ?

Solution

Example 3

. What are and ?

Solution

What about the two sided limit at ?

What do all the graphs have in common?

When the y value of a function approaches as x approaches a finite value , the graph of f will have a VERTICAL ASYMPTOTE.

The vertical line is a vertical asymptote for the function f , if any one of the following is true:

The TI 89 limit command

Try the limit command on the previous examples.

Limits at Infinity

Example 1

What are and ?

Solution

From the graph of the function , we see that and .

Confirm by looking at tables.

Example 2

What are and ?

Solution

Start by making a graph. In the figure below, the window is x = -10..1-, y=-2..2. Is it clear what the two limits are?

Here are some tables. Note how we've chosen our x values.

Both and are equal to 1.

Example 3

Let . What are and ?

Solution

The graph of P ( t ) on the window x = -6..6, y = -5..10

Tables

and .

Example 4

Let . Find and .

Solution

The graph of f ( x ) on the window x = -10..10, y = -4..4

Tables

does not exist.

What do all the graphs have in common?

When the y value of a function approaches a finite limit as , the graph of f will have a HORIZONTAL ASYMPTOTE.

The horizontal line is a horizontal asymptote for the function f , if aither of the following is true:

or .

The TI 89 limit command

Try the limit command on the previous examples.

Other examples

Find the following limits.

a.
b.
c.
d.
e.
f.
g
h
i.
j.