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**2.2 Limits Involving Infinity**

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**As the denominator gets larger, the value of the fraction gets smaller.**

There is a horizontal asymptote if: or

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**This number becomes insignificant as .**

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

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**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:

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Example 3: Find:

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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.

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**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….

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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

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**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.

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**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:

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**Right-end behavior models give us:**

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

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

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

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**Often you can just “think through” limits.**

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**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.

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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

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**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 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

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3.7 Graphing Rational Functions Obj: graph rational functions with asymptotes and holes and evaluate limits of rational functions.

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