Infinite Limits At Infinity



Infinite Limits At Infinity

Name ____________

Please read the following carefully.

Previously: Infinite Limits for Rational Functions

Given a rational function y = f(x), we were able to determine the end behaviors of the graph by using limits to calculate the asymptotes and to indicate the direction of the graph close to those asymptotes.

Vertical Asymptotes:

• Calculate vertical asymptotes by setting the ____________________ of a rational function equal to zero and solving for _________.

• Check the left and right side limits of the asymptotes to find out the behavior of the graph as it approaches the asymptotes.

• Ex) [pic]

Set denominator equal to zero

[pic]

Check limits on both sides of asymptotes to tell the direction the graph is going. Sub numbers on the left and right sides close to the value shown in the limit into each factor in the function. Fill in the blanks.

[pic]

These limits can be used to show the graph’s direction close to the asymptotes.

Horizontal Asymptotes:

• Remember the calculator method of finding horizontal asymptotes (this time, as x approaches infinity or gets very large, f(x) or y approaches some value)

1. Type equation into calculator ex)[pic]

2. Type 2nd WINDOW and change the table start to 100 000.

[pic]

3. Press 2nd GRAPH to see the horizontal asymptote shown in the Y1 column.

[pic]

Try one yourself:

Practice: Find the horizontal asymptote for [pic] using the calculator method. _________________

Now let’s look at the Calculus Method for finding horizontal asymptotes: We need to look at the limit as x gets very large or very small to figure out what y is doing.

• We find that calculating a limit as x approaches infinity can be shown by the symbols [pic]

• Remember the properties of infinite limits at infinity:

o [pic] When x is large, its reciprocal is very small and as x gets larger and larger, it follows that [pic] gets smaller and smaller and closer to zero.

o [pic] Likewise, when x is very large negative like – 1 000 000, [pic] is very small negative and also approaches zero.

o It follows that if x is raised to any exponent that the same pattern occurs so in general: [pic]

We can use this principle to find horizontal asymptotes by dividing every term in a rational function by the highest power of x to create and cancel out the basic infinite limits we see above.

• Consider this example where the rules from the previous page are used

Ex) [pic] Divide every term by the highest power of x.

[pic]

• Notice the relationship between the limit of the infinite limit at infinity and the horizontal asymptote.

Key: The value of the infinite limit is the horizontal asymptote.

So to find a horizontal asymptote, calculate the limit as x approaches infinity by dividing by the highest power of x.

Practice Questions:

1. Find the horizontal asymptote of

a. [pic].

b. [pic].

Key: To graph a rational function:

o Once the asymptotes and direction have been determined, plot the vertical asymptotes first, fill in the graph near them and then use the horizontal asymptote to complete the remainder of the graph.

Practice: Find the vertical and horizontal asymptotes and sketch the graph of the function.

a. [pic]

b. [pic]

Infinite Limits At Infinity on Polynomial Functions

We can use [pic] to show that as x gets very large so does f(x). If we are dealing with a polynomial function and not a rational function we use infinite limits at infinity to indicate the end behaviors of the graph.

For instance, consider the following limit:

a. [pic] As we sub in a sufficiently large negative value of x, we see that x2 approaches infinity.

Do the following yourself:

b. [pic] c. [pic] d. [pic]

Because we have seen the graphs of the above functions, we know that the end behaviors calculated by the limits are reasonable. If the function is more complex, it must be broken into factors before the limit test can occur.

Example: Find the end behavior of [pic].

a. Factor [pic]

b. Find the positive limit. (right side) [pic]

c. Find the negative limit. (left side) [pic]

With this information, the end behavior can be drawn to assist in the sketch of the graph.

Example: Find [pic].

[pic] [pic]

As x is sufficiently large positive, we see that all of the factors become large positive answers as well and the whole function approaches positive infinity.

Practice: Sketch the graph of y = (x-3)2(x+2)(1-x) by finding intercepts and infinite limits at infinity. (no calculators)

Does your graph look similar to this?

At the place where the arrow indicates, there is a touch rather than a cut. Explain why this is so.

Assignment:

• p. 222 #2(odds), 3(odds), 4(a, d-f, h), 5, 6

-----------------------

[pic]

We say the horizontal asymptote is y = 2.

These are the rules of infinite limits.

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