Clicker Question 1 What is the lim x  0- f (x ) for the function pictured on the board? A. 2 B. 0 C. -2 D. Does not exist.

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Presentation transcript:

Clicker Question 1 What is the lim x  0- f (x ) for the function pictured on the board? A. 2 B. 0 C. -2 D. Does not exist

Clicker Question 2 What is the lim x  0 f (x ) for the function pictured on the board? A. 2 B. 0 C. -2 D. Does not exist

Limits at Infinity and Global Asymptotes (2/6/09) By the “limit at infinity of a function f ″ we mean what f ′s value gets near as the input x goes out the positive (+  ) or negative (-  ) horizontal axis. We write lim x   f (x ) or lim x  -  f (x ). It’s possible that the answer can be a number, or be  or - , or not exist.

Examples lim x   1/(x + 4) = lim x   x + 4 = lim x  -  x + 4 = lim x   e x = lim x  -  e x = lim x   (2x +3)/(x – 1) = lim x   arctan(x ) =

Clicker Question 3 What is lim x   x / (x 2 +5) ? A. +  B. -  C. 0 D. 1 E. Does not exist

Clicker Question 4 What is lim x   x 2 / (x 2 +5) ? A. +  B. -  C. 0 D. 1 E. Does not exist

Clicker Question 5 What is lim x  -  x 3 / (x 2 +5) ? A. +  B. -  C. 0 D. 1 E. Does not exist

Nonexistent Limits at Infinity? Is it possible for a function to have no limit (including not +  nor -  )? If so, what is an example?

Global Asymptotes When lim x   f (x ) is a finite number a, then the graph of f has a horizontal asymptote, the line y = a. We can also call this a global asymptote since it describes the global (as opposed to local) behavior of f. But global asymptotes need not be horizontal lines nor even straight lines!

Examples f (x ) = x /(x – 2) has a horizontal global asymptote. What is it? g (x ) = x 2 / (x – 2) has a diagonal global asymptote. What is it? h (x ) = x 3 / (x – 2) has a parabolic global asymptote. What is it?

Assignment Monday we will have Lab #2 on power functions, polynomial functions, rational functions, and local and global behavior. Hand-in #1 is due at 4:45 on Tuesday. For Wednesday, please read Section 2.6 through page 137 and do Exercises 1, 3, 9, 15, 19, 28, 31, 35, 39 and 43.