Ms. Battaglia AB/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. x f(x) ? x approaches 2 from the left f(x) decreases without bound x approaches 2 from the right f(x) increases without bound

∞ 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 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.

Definition If 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.

Determine all vertical asymptotes of the graph of each function.

 Determine all vertical asymptotes of the graph.

 Find each limit.

Thm 1.15 Properties of Infinite Limits Let 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 -∞

Find each limit.

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| M. 1.The statementmeans that for each ε>0 there exits an N<0 such that |f(x)-L|<ε whenever x < N.

The line y=L is a horizontal asymptote of the graph of f if or Thm 3.10 Limits at Infinity If r is a positive rational number and c is any real number, then Furthermore, if x r is defined when x<0, then

 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 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.

 Find the limit

 Find each limit

Let 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. 1.The statement means that for each negative number M, there is a corresponding number N>0 such that f(x) N. Find each limit:

 Read 1.5 Page 88 #7, 9, 11, every other odd, 65, 68,  Read 3.5 Page 205 #1-6, odd, 90  Start preparing for Summer Material and Chapter 1 Test