Ch 3-1 Limits

THE LIMIT (L) OF A FUNCTION IS THE VALUE THE FUNCTION (y) APPROACHES AS THE VALUE OF (x) APPROACHES A GIVEN VALUE.

Easiest way to solve a limit Can’t use if it gives an undefined answer Direct Substituion Easiest way to solve a limit Can’t use if it gives an undefined answer

Table method and direct substitution method. because as x gets closer and closer to 2, x cubed gets closer and closer to 8. (graphically on next slide)

In this example it is fairly evident that the limit is 8 because when we replace x with 2 the function has a value of 8. This is not always as evident. Find the limit of as x approaches 1.

Rewrite before substituting Factor and cancel common factors – then do direct substitution. The answer is 4.

means x approaches a from the right and means x approaches a from the left

When finding the limit of a function it is important to let x approach a from both the right and left. If the same value of L is approached by the function then the limit exist and

THEOREM: As x approaches a, the limit of f (x) is L, if the limit from the left exists and the limit from the right exists and both limits are L. That is, if 1) and 2) then

Graphs can be used to determine the limit of a function Graphs can be used to determine the limit of a function. Find the following limits.

1.1 Limits: A Numerical and Graphical Approach a) Limit Graphically Observe on the graph that: 1) and 2) Therefore, does not exist.

1.1 Limits: A Numerical and Graphical Approach The “Wall” Method: As an alternative approach to Example 1, we can draw a “wall” at x = 1, as shown in blue on the following graphs. We then follow the curve from left to right with pencil until we hit the wall and mark the location with an × , assuming it can be determined. Then we follow the curve from right to left until we hit the wall and mark that location with an ×. If the locations are the same, we have a limit. Otherwise, the limit does not exist.

1.1 Limits: A Numerical and Graphical Approach Thus for Example 1: does not exist

1.1 Limits: A Numerical and Graphical Approach a) Limit Graphically Observe on the graph that: 1) and 2) Therefore,

We can also use Derive to evaluate limits.

Use Derive to find the indicated limit.

Limits at infinity Sometimes we will be concerned with the value of a function as the value of x increases without bound. These cases are referred to as limits at infinity and are denoted

For polynomial functions the limit will be + or - infinity as demonstrated by the end behavior of the leading term.

For rational functions, the limit at infinity is the same as the horizontal asymptote (y = L) of the function. Recall the method of finding the horizontal asymptote depends on the degrees of the numerator and denominator

1.1 Limits: A Numerical and Graphical Approach Limit Graphically Observe on the graph that, again, you can only approach ∞ from the left. Therefore, Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Find each limit

Infinite limits When considering Infinite limits When considering f(x) may increase or decrease without bound (becomes infinite) as x approaches a. In these cases the limit is infinite and

1.1 Limits: A Numerical and Graphical Approach b) Limit Graphically Observe on the graph that: 1) and 2) Therefore, does not exist.

When this occurs the line x = a is a vertical asymptote When this occurs the line x = a is a vertical asymptote. Polynomial functions do not have vertical asymptotes, but rational functions have vertical asymptotes at values of x that make the denominator = 0.

Find each limit.

The cost (in dollars) for manufacturing a particular videotape is where x is the number of tapes produced. The average cost per tape, denoted by is found by dividing C(x) by x. Find and interpret