2.2 Limits Involving Infinity

2.2 Limits Involving Infinity

As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if: or

This number becomes insignificant as . There is a horizontal asymptote at 1. Example 1:

Find: When we graph this function, the limit appears to be zero. so for : by the sandwich theorem: Example 2:

Example 3: Find:

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.

Example 4: The denominator is positive in both cases, so the limit is the same. Humm….

A function g is: a right end behavior model for f if and only if a left end behavior model for f if and only if End Behavior Models: End behavior models model the behavior of a function as x approaches infinity or negative infinity.

As , approaches zero. becomes a right-end behavior model. As , increases faster than x decreases, therefore is dominant. becomes a left-end behavior model. Example 5: (The x term dominates.) Test of model Our model is correct. Test of model Our model is correct.

becomes a right-end behavior model. On your calculator, graph: Use: becomes a left-end behavior model. Example 5:

Example 6: Right-end behavior models give us: dominant terms in numerator and denominator

Often you can just “think through” limits.

Definition of a Limit Let 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| < Ɛ Then we write Shortened version: If and only if for any number Ɛ >0, there is a real number ɗ >0 such that if x is within ɗ units of c (but x ≠ c), then f(x) is within Ɛ units of L.

3+Ɛ L=3 3-Ɛ ɗ is as large as possible. The graph just fits within the horizontal lines. → ← ɗ = 1/3 C=2

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) to • be between 8.9 and 9.1? • How close to 2 would you have to keep x in order for f(x) to • be within 0.001 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 small • Ɛ is. • What is the reason for the restriction “… but not equal to c” in • the definition of a limit? p

2.2 Limits Involving Infinity