OUR GOAL Find: Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS From sec 2.1 : OUR GOAL Find:

OUR GOAL Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS From sec 2.1 : OUR GOAL

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS For the function whose graph is given below Determine the following limits

Exam-I Term-122 Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Exam-I Term-122

1 2 Find: Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Find: 1 Use the Graph to find the limit 2 Create a table: watch the behavior

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Example: Find We say that the limit of f(x) as x approaches 2 from the left is equal to 4 We say that the limit of f(x) as x approaches 2 from the right is equal to 4 The symbol x  2- means that we consider only values of x < 2 The symbol x  2+ means that we consider only values of x > 2

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Example: Find

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Example: Example: Example: Example:

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Can we use Direct Substitution Prop.?? -

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Can we use Direct Substitution Prop.??

1 2 3 Find: Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Find: 1 Use the Graph to find the limit 2 Create a table: watch the behavior 3 Given the function f(x)

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS What is the difference ??

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Example: case 1)Common Factors 2)Conjugate radical Exam-1 Term 122 Exam-1 Term 121 Example

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS case 1)Common Factors 2)Conjugate radical 3)Open brackets

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Absolute Value Functions NOTE: 1) Study 2) In each side limit, use the definition of the absolute value

piecewise functions Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS piecewise functions NOTE: 1) Study 2) In each side limit, use the definition of the piecewise function

3 4 -1 1 4 Example: Example: Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS The Greatest Integer functions The largest integer that is less than or equal to x 3 4 -1 1 4 Example: Example: NOTE: 1) Study 2) In each side limit, substitute by a number

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

Squeeze theorem = Sandwich theorem Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Squeeze theorem = Sandwich theorem

Example: Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Example:

Exam1-091 Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Exam1-091

Postpone to sec2.6: Limits at Infinity: Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Postpone to sec2.6: Limits at Infinity: Limit from one side does not exist Find an example for a function such that the limit from right does not exist

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS

INFINITE LIMITS NOTE THAT: Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS INFINITE LIMITS NOTE THAT: The line x = 0 is a vertical asymptote

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS NOTE THAT: INFINITE LIMITS the line All red lines are ??????? Can we conclude? VA

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Example: Example: x y 3.1 10 3.001 1000 3.00001 100000 x y 2.9 10 2.999 1000 2.99999 100000 Note: Examples:

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Rational function (substitute by a) 1)Common Factors 2)Open brackets 3)…….. Direct substitution

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS Graph: y = ln x Graph: y = tan x is a vertical asymptote is a vertical asymptote

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS NOTE THAT: 1) Find the domain of f(x) 2) Study the limit 3) If the limit is + or - infinity

Exam-1 Term 152 Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS NOTE THAT: 1) Find the domain of f(x) 2) Study the limit 3) If the limit is + or - infinity Exam-1 Term 152

Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS