Section 19 - 1     Limits of Functions
Try the quiz at the bottom of the page!
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Our discussions in chapter 13 on limits were restricted to the usage of the natural numbers.  We are now ready to discuss limits in a broader sense.  We can now apply the concept of limits to all real numbers.  We can also extend the idea of a limit to include studying a limit as it approaches a finite number instead of always approaching one of the two infinities.  We will take a look at some examples of limits and also discuss some techniques for evaluating limits.  We will then relate the concept of limits to continuous/discontinuous functions!  Hang on to your hat!
Simple limits on infinity.
Evaluate each of the following limits:
       Using your calculator, you will get the following table:
x 1 8 27 64
y 1 2 3 4
 
As you study the y values which represent the limit, you can see that the numbers are getting arbitrarily big.  Therefore, the limit is positive infinity.
          Using your calculator, you will again get the following table:
x -1 -8 -27 -64
y -1 -2 -3 -4
Again, looking at the y values, leads you to conclude that as you take larger and larger negative numbers, the limit approaches negative infinity.
The limit as x approaches some real number c!
Study the graph to illustrate the above two limits!
Use the graph to find each:  and  .  To find the first limit, follow the graph from the far left until that path comes to an end.  What y value does it approach?  From the graph, it approaches 3!  Now for the second limit.  Follow the graph from the far right until the path comes to an end. This one approaches the y value -2!  Notice, that neither limit approached the actual value of f(2) which is 1!  Also note that the limit as x approaches 2 does not exist! Why?  Both limits do not approach the same value!!  Remember the 3 possibilities from chapter 13?
Quotient theorem
If the lim f(x) and lim g(x) both exist and lim g(x)  0, then:
This theorem helps find many limits but it must be in the correct form.  Notice, that both limits must exist and lim g(x)  0.  Then to find the limit is easy.  It's the limit of the numerator over the limit of the denominator!
Unfortunately, not many limits can be solved using the above formula.  Most limits don't fit neatly into the correct form.  So you might find the following helpful in determining limits.
Techniques for evaluating:  
Sample problems
Continuous vs. Discontinuous Functions
Formal definition of continuous:
A function is continuous at a real number c if:
  = f(c) provided these 3 conditions are met:
The following graphs give examples of discontinuous functions:
 
 
Current quizaroo #  19a
 
 Find each of the following limits!!
 
a) 0
b) 1/9
c) infinity
d) negative infinity
e) limit doesn't exist
 
a) -14
b) 0
c) infinity
d) negative infinity
e) does not exist
 
 
a) -1/2
b) 1/2
c) 0
d) -1/4
e) 1/4
 
a) 1/2
b) -1/2
c) 0
d) infinity
e) negative infinity
 
a) 1
b) -1
c) 0
d) infinity
e) negative infinity
 
 

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