AP CALCULUS Limits 1: Local Behavior

REVIEW: ALGEBRA is a ________________________ machine that ___________________ a function ___________ a point. CALCULUS is a ________________________ machine that ___________________________ a function ___________ a point

Limits Review: PART 1: LOCAL BEHAVIOR (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

Analytically Find a if given and for

Find a for any

Day 2

FINDING LIMITS

G N A W X Mantra: Numerically Words Verify these also:

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

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 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

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

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

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 f g h a This theorem allow us to use DIRECT SUBSTIUTION with Trig Functions.

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

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