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

12.2 Techniques for Evaluating Limits

There are many techniques for finding the limit of a function. Some of these techniques involve algebra, while others involve using a calculator.  Some of the different techniques we will look at are as follows:

1. Direct Substitution
2. Dividing Out
3. Rationalizing

We will also look into:
1. Using Calculators
2. Evaluating One Sided Limits
3. Evaluating Limits from Calculus

Direct Substitution

The first technique for evaluating limits is by using Direct Substitution. This method involves substituting the number theta into your function. For example:







So in this case, the limit of this function is -4. However, the situation changes if the function you begin with is a fraction. This is because if you use the direct substitution method, you can get zeros for your limits which is not correct. In this case, you would need to use the next method: Dividing Out.

Dividing Out

 This technique is very useful when dealing with polynomials. In this method, you will use the fraction to your advantage by dividing out some of the zeros. Here is an example:

 Because we could factor the polynomial in the numerator, we were able to have a common factor that divided out. This left us with a simple subtraction problem where all that we needed to do is plug in for x and solve. If you are given a less friendly equation to begin with, another method used to find limits is called the rationalizing technique.

Rationalizing

This method may be helpful if you are given a problem with a root. To start the rationalizing process, you must multiply both the numerator and the denominator by the conjugate. After this, simplify the equation, and then use the direct substitution method to evaluate the limit. Example:

 Now that you know the limit is not 0, you can proceed to use the direct substitution method...


 

The limit of this function is 1/6. As you can see, rationalizing is a good way to evaluate a limit when you have a root involved.


Using Calculators

Using a calculator can sometimes be a very fast way to deal with limits. The calculator enables you to make a graph of the function, see the table from that graph, or even make your own table. If the direct substitution method produces an indeterminate limit, the graphing calculator will give the correct limit.


Evaluating One Sided Limits

Here is an example of a one sided limit:

For this scenario you must evaluate the limit from both sides of the graph, coming from the right and the left. If you are evaluating the right side of the graph, the limit notation will have a small + sign, and from the left it will have a small - sign. To evaluate this limit, you use the same methods as before, but you must evaluate both sides of the graph.

Evaluating Limits in Calculus

Unfortunately, the fun with limits continues in Calculus :(  It may be seen while using the difference quotient, like this example:

 

Have fun with limits. Wow it's late. I think I'll go to bed now.

12.1 Intro to Limits

Definition of Limit

If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L.

Official Notation:

Simplified Definition:

As x get closer to a point from both side, what would the y or f(x) value most likely be without considering what it actually is.

The limit is what the approaching numbers imply, NOT the actual value.

Solve by Graphing:

Function  

Limit

By looking at the graph, as x approaches the value 1, the line gets closer to the y value 6.

Answer

Types of limits

A limit can either exist or not exist with a graph. Existing one may look a lot like the one above with a few varieties. Non-existing limits can be separated into two categories, one that don't exist and infinities.

A limit as x approaches 2 wouldn't exist.
Either side of the limit approach different numbers.

A limit at 1 could be two things.
(depending on strictness of definition)
the answer could be infinity or does not exist.

With limits that have not definite infinity, they lye on the line between existence. The definition says that it is a number, and since infinity is not a single number, it doesn't exist. But infinity is also true and is useful in calculus.

Solve with a Table:

This process involves plugging in number extremely close to the limit. If both sides of the limit get extremely close to a number, that number is the limit.