Limits Approaching Positive and Negative Infinity

, The value of a limit as x approaches positive or negative infinity, also known as End Behavior, can be one of three things:  positive or negative infinity, a real number, or Does Not Exist.

Example 1:

if, what is ?

As the value of x approaches , what is the value of f(x)?

To find out, there are two methods: graphing and using tables, and substitution. 

The graph of f(x) shows that as x increases, so does f(x), and as x grows infinitely large, so will f(x). 

Common sense at this point would have the value of  be equal to . But this is Pre-Calculus, common sense flew out the window the first day of first trimester. So to show that as x approaches  so does f(x), we must make a table.

1	10	100	1000
3	102	10002	1000002

As you can see, as x approaches f(x) approaches infinty. If we subsititute infinity for x, we have the equation simplifying, we get

so, = 

Example 2:

 Now that the easy stuff is out of the way, lets do a limit that's worthy of Honor's Pre-Calculus.

if  , what is?

If we look at the graph, 

 We can see that the equation never approaches -, so what happens to f(x) when x approaches -? If we substitute - for x,we get. Any root of a negative number is not real. There are no real values of f(x) as x approaches infinity, so Does Not Exist (DNE)

Example 3:

But what about when the limit is a real number? 

If  , what is ?

If we look at the graph

We can see that as x increases, it gets closer and closer to a number that is somewhere between 2 and 3.

But, what do we need to do to get the exact value? SUBSTITUTE!

=0, therefore

But, why isn't 1?  Because x never reaches , it just approaches it. In order to see what happens to f(x) as x approaches , just substitute a large number for x and make a table.

1	10	100	1000
3	2.5937	2.7048	2.7169

As x approaches  , f(x) gets closer and closer to 2.72

So, is  2.72