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) Using Limits: A vertical asymptote of x = a exists for a function if OR

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

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:

Special Limits 0 0 Therefore, by the Sandwich Theorem,

Special Limits Example: What is

Special Limits Example: What is

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)

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.

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  -∞

End Behavior Show that g(x) = 3x 4 is an end behavior model for f(x) = 3x 4 – 2x 3 + 3x 2 – 5x + 6.

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) = x is a right hand behavior model for f(x)

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

Finding End Behavior Models Find a left end behavior model for the function f(x) = x + e  x 01 Therefore, e –x is a left-end behavior model for f(x).

HW Section 2.2 (#1-7 odd,, 21, 23, 25, odd, 39, 41, 43, odd) Web Assign due Monday night