Section 2.2 The Limit of a Function AP Calculus September 10, 2009 CASA

Calculus, Section 2.22 “Calculus Close” In calculus, we will continually look at what happens in very small intervals. The book uses the term “arbitrarily close” to describe a situation where we are close enough to see what is at a point, but not actually at that point Mr. Pierce, calculus teacher at Buffalo Grove High School (IL), calls this “calculus close”

Calculus, Section 2.23 Definition of Limit “The limit of f of x as x approaches a.” The limit of a function is the value of the function as it approaches, but does not reach it’s destination.

Calculus, Section 2.24 Situation 1 Situation 1: Function is continuous at the point in question. The limit of f(x) as x approaches 2 is 4 Also, f(2)=4 (The function is continuous at 2)

Calculus, Section 2.25 Situation 2 Function has a hole (the function is not continuous) and both sides of the function approaches the empty spot.

Calculus, Section 2.26 Situation 3 The function is not continuous, both sides of the function does not approach the hole. We say the limit exists when approaching from the left (-), exists when approaching from the right (+), but the limit DOES NOT EXIST (DNE)

Calculus, Section 2.27 Situation 4.1 When function has a vertical asymptote, we say the “limit of the function approaches infinity.” a b c

Calculus, Section 2.28 Situation 4.2 If the left-hand limit does not agree with the right-hand limit, we say the limit does not exist. a b c

Calculus, Section 2.29 Situation 4.3 When function has a vertical asymptote, we say the “limit of the function approaches infinity.” a b c

Calculus, Section Definition of a Vertical Asymptote If has a vertical asymptote at at least one of these statements is true

Calculus, Section Using the TI-83 (trends)

Calculus, Section Using the TI-83 (warnings)

Calculus, Section Using the TI-83 (warnings)

Calculus, Section Assignment Section 2.2, 1-39 odd