limits of functions

Section 19 - 1 Limits of Functions

Try the quiz at the bottom of the page! go to quiz 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!

1) , means the limit of f(x) as x approaches c from the right!

, means the limit of f(x) as x approaches c from the left!

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:

1) If possible, use the quotient theorem. Check this first. Couldn't hurt!

2) If lim f(x) = 0 and lim g(x) = 0, try one of the following:

a) Factor f(x) and g(x) and reduce to lowest terms.

b) If f(x) or g(x) involves a square root, multiply both the numerator and the denominator by the conjugate of the square root expression.

3) If lim f(x) 0 and lim g(x) = 0 then one of the following is true:

a) or

b) lim doesn't exist.

4) If x approaches infinity or negative infinity, divide the numerator and the denominator by the highest power of x in the denominator. (This sounds familiar!)

5) If all else fails, get out the calculator and crunch the numbers. If x approaches infinity or negative infinity use very large or very small values and watch the pattern develop. If the limit approaches a number c, then use numbers close to c on both sides of c.

Sample problems

1) Because the problem has a root in it we will try 2b.
2) Because the lim of the num and den = 0, try 2a.
3) Because the limit of the num and dem both exist, you can use 1.
4) Because the lim f(x) is not zero, and the lim g(x) = 0, it must be choice 3.

x	.1	.01	.001	-.1	-.01	-.001
y	11	101	1001	-9	-99	-999

From the chart you can see that as the numbers get closer to 0 from the right, the limit is infinity, but as you approach from the left, the limit is negative infinity. Therefore, the limit does not exist!
5) Because the limit goes toward infinity it's #4.

Continuous vs. Discontinuous Functions

Basically, a continuous function is a function you can draw without lifting your pencil. There are no breaks in the graph or holes in the graph. But to say that a function that is defined for all values of x makes it a continuous function is not accurate. There are many functions that have the real numbers as their domain and the function is not continuous. A more formal definition is as follows:

Formal definition of continuous: A function is continuous at a real number c if: