6Infinite Limits:As the denominator approaches zero, the value of the fraction gets very large.vertical asymptote at x=0.If the denominator is positive then the fraction is positive.If the denominator is negative then the fraction is negative.
7The denominator is positive in both cases, so the limit is the same. Example 4:The denominator is positive in both cases, so the limit is the same.Humm….
8End Behavior Models:End behavior models model the behavior of a function as x approaches infinity or negative infinity.A function g is:a right end behavior model for f if and only ifa left end behavior model for f if and only if
9becomes a right-end behavior model. Example 5:As , approaches zero.(The x term dominates.)becomes a right-end behavior model.Test ofmodelOur modelis correct.As , increases faster than x decreases,therefore is dominant.becomes a left-end behavior model.Test ofmodelOur modelis correct.
10becomes a right-end behavior model. Example 5:becomes a right-end behavior model.becomes a left-end behavior model.On your calculator, graph:Use:
11Right-end behavior models give us: Example 6:Right-end behavior models give us:dominant terms in numerator and denominator
12Example 7:Right-end behavior models give us:dominant terms in numerator and denominator
13Example 8:Right-end behavior models give us:dominant terms in numerator and denominator
150 < |x - c|< ɗ such that |f(x) – L| < Ɛ Definition of a LimitLet c and L be real numbers. The function f has limit L as x approaches c (x≠c), if, given any positive Ɛ, there is a positive number ɗ such that for all x, if x is within ɗ units of c, then f(x) is within Ɛ units of L.0 < |x - c|< ɗ such that |f(x) – L| < ƐThen we writeShortened version:If and only if for any number Ɛ >0, there is a real number ɗ >0such that if x is within ɗ units of c (but x ≠ c), then f(x) is withinƐ units of L.
163+ƐL=33-Ɛ→← ɗ = 1/3ɗ is as large as possible. The graph just fits within the horizontal lines.C=2
17p Plot the graph of f(x). Use a friendly window that includes x = 2 as a grid point. Name the feature present at x = 2.From the graph, what is the limit of f(x) as x approaches 2.What happens if you substitute x = 2 into the function?Factor f(x). What is the value of f(2)?How close to 2 would you have to keep x in order for f(x) tobe between 8.9 and 9.1?be within unit of the limit in part 2? Answer in the form“x must be within ____ units of 2”What are the values of L, c, Ɛ and ɗ?Explain how you could find a suitable ɗ no matter how smallƐ is.What is the reason for the restriction “… but not equal to c” inthe definition of a limit?p