Welcome to Class Arrival Instructions: Factor each of the following functions.

Evaluating Limits Algebraically Day 3 Evaluating Limits Algebraically

Algebraic Techniques Direct substitution Factoring Rationalizing roots Multiply by 1 Synthetic Division A few to memorize Sometimes the limit does not exist

Based on yesterday’s notes: Is often independent of f(c) However, for continuous functions We will refer to this as DIRECT SUBSTITUTION to find a limit.

Example

DIRECT SUBSTITUTION can be used if. . . The function is continuous at c. Which means (informally): No holes No jumps No vertical asymptotes

f(-4) D.N.E., however if we factor we can find Dealing with “holes” . . . . f(-4) D.N.E., however if we factor we can find

Practice:

Dealing with zero Undefined and indeterminate are NOT the same thing! Undefined is an answer and means it does not exist. Indeterminate means we do not know the answer…yet.

“Indeterminate Form”: when direct substitution produces 0/0 Remember Conjugates?!?...

You try:

Back in Time Fractions LCDs (Least Common Denominators) To get rid of the denominators, multiply by . . .

Multiply by 1 in a “convenient form” (The common denominator)

Synthetic Division

KNOW THESE!!

Together Let’s Try

Summary Direct substitution Factoring Rationalizing roots Multiply by 1 Synthetic Division A few to memorize Sometimes the limit does not exist

One last way to find a limit… Squeeze Thm (aka Sandwich Thm): If and then

Sandwich Thm Example