1.5/3.5 Infinite LimitsObjective: Determine Infinite Limits from the left and

right; determine horizontal asymptotes.

Ms. BattagliaAB/BC Calculus

Let f be the function given by 3/(x-2)

A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite limit.

Infinite Limits

x 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5f(x) -6 -30 -300 -3000 ? 3000 300 30 6

x approaches 2 from the left

f(x) decreases without bound

x approaches 2 from the right

f(x) increases without bound

Definition of Infinite Limits ∞

Let f be a function that is defined at every real number in some open interval containing c (except possibly c itself). The statement

means that for each M>0 there exists a δ>0 such that f(x)>M whenever 0<|x-c|<δ. Similarly,

means that for each N<0 there exists a δ>0 such that f(x)<N whenever o<|x-c|<δ.

To define the infinite limit from the left, replace 0<|x-c|<δ by c-δ<x<c. To define the infinite limit from the right, replace 0<|x-c|<δ by c<x<c+δ

Determine the limit of each function shown as x approaches 1 from the left and from the right.

Determining Infinite Limits from a Graph

Vertical AsymptoteDefinitionIf f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x=c is a vertical asymptote of the graph of f.Thm 1.14 Vertical Asymptotes

Let f and g be continuous on an open interval containing c. If f(c)≠0, g(c)=0, and there exists on an open interval containing c such that g(x) ≠0 for all x≠c in the interval, then the graph of the function given by

has a vertical asymptote.

Finding Vertical Asymptotes

Determine all vertical asymptotes of the graph of each function.

A Rational Function with Common Factors Determine all vertical asymptotes of the graph.

Determining Infinite Limits Find each limit.

Thm 1.15 Properties of Infinite LimitsLet c and L be real numbers and let f and g be functions such that

1. Sum or difference:

2. Product: , L > 0

, L<0

3. Quotient:

Similar properties hold for one-sided limits and for functions for which the limit of f(x) as x approaches c is -∞

Determining LimitsFind each limit.

Definition of Limits at Infinity

Let L be a real number.

1.The statement means that for each ε>0 there exists an M>0 such that |f(x)-L|<ε whenever x>M.

2.The statement means that for each ε>0 there exits an N<0 such that |f(x)-L|<ε whenever x < N.

Horizontal AsymptoteThe line y=L is a horizontal asymptote of the graph of f if

or

Thm 3.10 Limits at InfinityIf r is a positive rational number and c is any real number, then

Furthermore, if xr is defined when x<0, then

Find the limit

Find the limit

Find each limit

1. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0.

2. If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients.

3. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function does not exist.

Guidelines for Finding Limits at + of Rational Functions

Find the limit

Find each limit

Definition of Infinite Limits at InfinityLet f be a function defined on the interval (a,∞)

1. The statement means that for each positive number M, there is a corresponding number N>0 such that f(x)>M whenever x>N.

2. The statement means that for each negative number M, there is a corresponding number N>0 such that f(x)<M whenever x>N.

Find each limit:

Read 1.5 Page 88 #7, 9, 11, 21-49 every other odd, 65, 68, 73-76

Read 3.5 Page 205 #1-6, 19-33 odd, 90 Start preparing for Summer Material and Chapter 1 Test

Classwork/ Homework