6 Infinite 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.
7 The 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….
8 End 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
9 becomes 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.
10 becomes a right-end behavior model. Example 5:becomes a right-end behavior model.becomes a left-end behavior model.On your calculator, graph:Use:
11 Right-end behavior models give us: Example 6:Right-end behavior models give us:dominant terms in numerator and denominator
12 Example 7:Right-end behavior models give us:dominant terms in numerator and denominator
13 Example 8:Right-end behavior models give us:dominant terms in numerator and denominator
15 0 < |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.
16 3+ƐL=33-Ɛ→← ɗ = 1/3ɗ is as large as possible. The graph just fits within the horizontal lines.C=2
17 p 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