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

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

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

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

5. Finding Limits Algebraically
The most effective way of finding a limit algebraically is DIRECT SUBSTITUTION.
DIRECT SUBSTITUTION
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.

6. 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.
The limit of a difference is the difference of the limits.
The limit of a product is the product of the limits.

7. 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.
The limit of a scalar times a function is the scalar
times the limit of the function.
Example Problems

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

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

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

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