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

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

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.

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….

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 if a left end behavior model for f if and only if

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

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:

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

Example 7: Right-end behavior models give us: dominant terms in numerator and denominator

Example 8: Right-end behavior models give us: dominant terms in numerator and denominator

Often you can just “think through” limits.

0 < |x - c|< ɗ such that |f(x) – L| < Ɛ 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 = 1/3 ɗ is as large as possible. The graph just fits within the horizontal lines. C=2

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) to be between 8.9 and 9.1? 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