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2.2 Limits Involving Infinity

2.2 Limits Involving Infinity. Graphically. What is happening in the graph below?. Graphically. We can make the following statements:. ALSO:. Vertical Asymptotes. When do vertical asymptotes occur algebraically?. Denominator = 0 (a function is undefined…this includes trig functions).

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2.2 Limits Involving Infinity

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  1. 2.2Limits Involving Infinity

  2. Graphically • What is happening in the graph below?

  3. Graphically We can make the following statements: ALSO:

  4. Vertical Asymptotes • When do vertical asymptotes occur algebraically? • Denominator = 0 • (a function is undefined…this includes trig functions) • Using Limits: A vertical asymptote of x = a exists for a function if OR

  5. Horizontal Asymptotes • A horizontal asymptote of y = b exists if OR Example: Identify all horizontal and vertical asymptotes of

  6. Special Limits Example: What is If we substitute in ∞, sin ∞ oscillates between -1 and 1, so we must find another way to show this limit algebraically. USING SANDWICH THEOREM:

  7. Special Limits 0 0 Therefore, by the Sandwich Theorem,

  8. Special Limits Example: What is

  9. Special Limits Example: What is

  10. Limits Involving ±∞ • The same properties of adding, subtracting, multiplying, dividing, constant multiplying, and using powers for limit also apply to limits involving infinity. (see pg. 71)

  11. End Behavior • We sometimes want to how the ends of functions are behaving. • We can use much simpler functions to discuss end behavior than a complicated one that may be given. • To look at end behavior, we must use limits involving infinity.

  12. End Behavior A function g is an end behavior model for f if and only if Right-end behavior model when x +∞ Left-end behavior model when x -∞

  13. End Behavior • Show that g(x) = 3x4 is an end behavior model for f(x) = 3x4 – 2x3 + 3x2 – 5x + 6.

  14. Finding End Behavior Models Find a right end behavior model for the function f(x) = x + ex Notice when x is ∞, e∞goes to 0. If we use a function of g(x) = x in the denominator, we get 0 Therefore, g(x) = xis a right hand behavior model for f(x)

  15. Finding End Behavior Models Find a left end behavior model for the function f(x) = x + ex Notice when x is ∞, exgoes to ∞ and x goes to –∞. Which one has more effect on the left-end of the function? (Which one gets to ∞ faster?) e∞ Therefore, use e–x as a left-end behavior model for f(x).

  16. Finding End Behavior Models Find a left end behavior model for the function f(x) = x + ex 0 1 Therefore, e–x is a left-end behavior model for f(x).

  17. HW • Section 2.2 (#1-7 odd, , 21, 23, 25, 27-33 odd, 39, 41, 43, 45-49 odd) • Web Assign due Monday night

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