AP CALCULUS Limits 1: Local Behavior

You have 5 minutes to read a paragraph out of the provided magazine and write a thesis statement regarding what you read

C CONVERSATION: Voice level 0. No talking! H HELP: Raise your hand and wait to be called on. A ACTIVITY: Whole class instruction; students in seats. M MOVEMENT: Remain in seat during instruction. P PARTICIPATION: Look at teacher or materials being discussed. Raise hand to contribute; respond to questions, write or perform other actions as directed. NO SLEEPING OR PUTTING HEAD DOWN, TEXTING, DOING OTHER WORK. S Activity: Teacher-Directed Instruction

Objectives(SWBAT): Content: evaluate limits using basic limit laws, direct substitution, factoring, and rationalizing Language: SW verbally describe limit laws in their own words

REVIEW: ALGEBRA is a _________________ machine that ___________________ a function ___________ a point. CALCULUS is a ________________________ machine that ___________________________ a function ___________ a point function evaluates Limit Describes the behavior of near

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) (3). Notation (4). Mantra x  a y  L L a

G N A W Graphically “We Don’t Care” Postulate”: What is the y value? 0 3

G N A W Numerically x y err or 40.2

(5). Formal Definition ( Equation Part) Graphically: Find a If

Analytically Find a if given and for

Find a for any

C CONVERSATION: Voice level 0. No talking! H HELP: Raise your hand and wait to be called on. A ACTIVITY: Whole class instruction; students in seats. M MOVEMENT: Remain in seat during instruction. P PARTICIPATION: Look at teacher or materials being discussed. Raise hand to contribute; respond to questions, write or perform other actions as directed. NO SLEEPING OR PUTTING HEAD DOWN, TEXTING, DOING OTHER WORK. S Activity: Teacher-Directed Instruction

Objectives(SWBAT): Content: evaluate limits using basic limit laws, direct substitution, factoring, and rationalizing Language: SW verbally describe limit laws in their own words

G – Graphically N – Numerically A – Analytically W -- Words

The Formal Definition Layman’s definition of a limit As x approaches a from both sides (but x≠a) If f(x) approaches a single # L then L is the limit

FINDING LIMITS

G N A W X Mantra: Numerically Words Verify these also: x  a, y  L Must write every time

(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 Direct substitution 13 Indeterminate form

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 ! Creates a hole so you either factor or rationalize

Do All Functions have Limits? Where LIMITS fail to exist. Why? f(x) approaches two different numbers Approaches ∞Oscillates At an endpoint not coming from both sides

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