1002 Limits 1 Local Behavior AP CALCULUS REVIEW

1002 - Limits 1: Local Behavior AP CALCULUS

REVIEW: ALGEBRA is a ____________ machine that __________ a function ______ a point. CALCULUS is a ____________ machine that ______________ a function ______ a point

PART 1: LOCAL BEHAVIOR Limits Review: (1). General Idea: Behavior of a function very near the point where (2). Layman’s Description of Limit (Local Behavior) L a (3). Notation (4). Mantra

G N A W Graphically “We Don’t Care” Postulate”:

G N A W Numerically

The Formal Definition

(5). Formal Definition ( Equation Part) Graphically: Find a If 3 2 1 2 3 4

• Analytically Find a if given and for ---------------------

Find a for any

Day 2

FINDING LIMITS

G N A W • Numerically -. 1 -. 001 0. 001. 01 X • Words Mantra: Verify these also: . 1

(6). FINDING LIMITS • Graphically “We Don’t Care” Postulate…. . • The existence or non-existence of f(x) at x = 2 has no bearing on the limit as

FINDING LIMITS • Analytically A. “a” in the Domain Use ________________ B. “a” not in the Domain This produces ______ called the ___________ Rem: Always start with Direct Substitution

Method 1: Algebraic - Factorization Rem: Always start with Direct Substitution Method 2: Algebraic - Rationalization Method 3: Numeric – Chart (last resort!) Method 4: Calculus To be Learned Later !

Do All Functions have Limits? Where LIMITS fail to exist. Why?

Review : 1) Write the Layman’s description of a Limit. 2) Write the formal definition. ( equation part) 3) Find each limit. 4) Does f(x) reach L at either point in #3?

Homework Problems 1. From the figure, determine a such that

Review: (5). The graph of the function displays the graph of a function with Estimate how close x must be to 2 in order to insure that f(x) is within 0. 5 of 4. (6). Find a such that

Last Update: • 08/12/10

Properties of Limits Using Direct Substitution BASIC (k is a constant. x is a variable) 1) 2) 3) 4) IMPORTANT: Goes BOTH ways!

Properties of Limits: cont. OPERATIONS Take the limits of each part and then perform the operations. EX: POLYNOMIAL, RADICAL, and RATIONAL FUNCTIONS all us Direct Substitution as long as a is in the domain

Composite Functions REM: Notation THEOREM: and Use Direct Substitution. EX:

Limits of TRIG Functions Squeeze Theorem: if f(x) ≤ g(x) ≤ h(x) for x in the interval about a, except possibly at a and the Then exists and also equals L h g a f This theorem allow us to use DIRECT SUBSTIUTION with Trig Functions.

Limits of TRIG Functions: cont. In a UNIT CIRCLE measured in RADIANS: Defn. of radians! THEREFORE:

Exponential and Logarithmic Limits Use DIRECT SUBSTITUTION. REM: the Domain of the functions For a > 0 REM: Special Exponential Limit