Finding Limits An Algebraic Approach AP Calculus August

Finding Limits: An Algebraic Approach AP Calculus August 31, 2016 Mrs. Agnew

Essential Questions n n How do you find the limit of a function algebraically? When does a limit fail to exist? Essential Vocabulary n n n Unbounded limit Direct Substitution Limit Laws

Limits that Fail to Exist… There are three conditions under which a limit fails to exist. 1. 2. 3. Limit from right ≠ limit from left. Limit is unbounded (approaches ∞ or -∞). Limit oscillates between two values.

Finding Limits Numerically n To find a limit numerically means to choose values for x close to a on either side of a. n We will investigate what happens to the y values as x approaches a. q n Use the table feature on your calculator. Examples…

Finding Limits Algebraically n The most effective way of finding a limit algebraically is DIRECT SUBSTITUTION. n DIRECT SUBSTITUTION q q n Plug in value for x Does not work if it produces undefined value. Direct substitution works if limit of function is equal to value of function at x = a.

The Limit Laws Assume the limits of f(x) and g(x) as x → a exist. The limit of a sum is the sum of the limits. 1. The limit of a difference is the difference of the limits. 2. The limit of a product is the product of the limits. 3.

The Limit Laws Assume the limits of f(x) and g(x) as x → a exist. The limit of a quotient is the quotient of the limits. 4. 5. The limit of a scalar times a function is the scalar times the limit of the function. Example Problems

Evaluating Limits Algebraically n Methods of finding limits (so far): q q n Numerically and Graphically Direct Substitution Direct substitution works as long as the limit of the function equals the value of the function. q Plug in value for x to find limit. q What if we get an undefined value when we plug in the value for x?

Evaluating Limits Algebraically n Consider the function below. n Find the limit as x → 3. n Could we use direct substitution to find the limit? q But just because direct substitution does not work, that does not mean that the limit

Finding Limits Algebraically When direct substitution does not work, n you may need to manipulate the function to use direct substitution. q q q n Factor and Cancel Rationalize (numerator or denominator) Eliminate Compound Fractions Example Problems

Homework: August 31, 2016 Page 67 – 69 #19, 21, 33 – 35, 37 – 40 47 – 61 (Odd), 64, 103