Limits at Infinity. as x, f (x)?

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Cameron Parker
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1 Limits at Infinity as x, f (x)? as x, f (x)?

2 Let s look at...

3 Let s look at...

4 Let s look at...

5 Definition of a Horizontal Asymptote: If Then the line y = L is called a horizontal asymptote of the graph of f. (so, at most, a function can have 2 horizontal asymptotes)

6 Limits at infinity have many of the same properties of limits discussed in Section 2.1 (pages 58-60)

7 Limits at infinity have many of the same properties of limits discussed in Section 2.1 (pages 58-60)

8 Theorem 4.10 Limits at Infinity If r is a positive rational number and c is any real number, then Furthermore, if is defined when x < 0, then

9 Applying the rules of limits & Theorem 4.10

10 Applying the rules of limits & Theorem 4.10

11 To determine the limit at infinity for a rational function (polynomial/ polynomial).. one common method is to divide every term by the largest power of x in the DENOMINATOR, then evaluate the limit of the result

12 Examples:

13 Examples: dividing every term by largest power in denominator

14 Two more examples, then some principles:

15 Two more examples, then some principles:

16 You do.

17 You do.

18 General principles for limits at infinity for rational functions If the highest power is in the denominator, the function approaches 0 as x approaches infinity or negative infinity If the highest power is in the numerator, the function grows without bound (some would say the limit is infinity). If degree is just one greater in numerator than denominator there is a SLANT ASYMPTOTE. If the numerator and denominator have equal high powers, then the function approaches the coefficients on the largest power

19 Word problem: Suppose that f (t) measures the level of oxygen in a pond, where f (t) = 1 (100%) is the normal (unpolluted) level and the time, t, is measured in weeks. When t = 0, organic waste is dumped into the pond, and as the waste material oxidizes, the amount of oxygen in the pond is given by... What percentage of the normal level of oxygen exists in the pond after 1 week? After 10 weeks? As t approaches infinity?

20 What percentage of the normal level of oxygen exists in the pond after 1 week? After 10 weeks? As t approaches infinity?

21 A closer look at Slant Asymptotes If the highest power is in the numerator, the function grows without bound (some would say the limit is infinity). If degree is just one greater in numerator than denominator there is a SLANT ASYMPTOTE. Let s look at some examples.

22 Since the largest power in the numerator is greater than that of the denominator we know.. Since the degree of the numerator is one greater than that of the denominator, we a slant asymptote describes the growth We ll do the polynomial division to find the slant asymptote.

23 The root is 4, so that s what we divide by Disregard the remainder, the slant asymptote s equation is f(x)= x - 2

24 Another.. We know Let s find the slant asymptote Disregard the remainder.the equation of the slant asymptote is f(x)=x

25 For you to do. Find the limit and the slant asymptote.. Polynomial Division: By Synthetic Division:

26 The Graph..

27 Exploration Activity on graphs: Describe the important features of the graph. Can a single viewing window show all the features clearly? What are the horizontal asymptotes?

28 Describe the important features of the graph. Closer looks at the 2 vertical asymptotes.

29 Describe the important features of the graph. Vertical asymptotes: x-values that make the denominator = 0

30 Describe the important features of the graph. Zero s of function where numerator = 0 and the denominator does not equal zero

31 Can a single viewing window show all the features clearly? Maybe, but it is better to change viewing windows. Confirm values by going to the table. (1) Know Range&Domain (2) Intercepts/Asymptotes (3) Critical numbers/inflection points What are the horizontal asymptotes? Highest degree same on the top and bottom, so horizontal asymptote is 2/3

32 Problems for you.... From page 188, #1-8 (matching) & #9-23 odd problems

Math 121. Graphing Rational Functions Fall 2016

Math 121. Graphing Rational Functions Fall 2016 1. Let x2 85 x 2 70. (a) State the domain of f, and simplify f if possible. (b) Find equations for the vertical asymptotes for the graph of f. (c) For each