1.3 Evaluating Limits Analytically Objectives: -Students will evaluate a limit using properties of limits -Students will develop and use a strategy for finding limits -Students will evaluate a limit using dividing out and rationalizing techniques -Students will evaluate a limit using the Squeeze Theorem

Properties of Limits

Ex 1)

Ex 2) Yesterday we found that but we had to do this graphically; direct substitution didn’t work because of the hole (the denom was 0) To find the limit analytically…simplify! Find the same function but without the hole.

Strategy for finding : 1)Direct substitution 2)Simplify to an identical function except at x=c ; then use direct substitution. 3)Use a graph or table to check. 4)Remember, sometimes the limit DNE!!

Simplifying Techniques: 1) Factoring (cancellation)

Simplifying Techniques: 2) Rationalization (rationalize numerator by multiplying by conjugate over itself)

Simplifying Techniques: 3) ∆x problems → expand then reduce

Simplifying Techniques: 4) Common denominator

Other Things to Look For: Squeeze Theorem -if for all x in an open interval containing c, except possibly at c, and if, then

Two Special Trigonometric Limits: ** memorize these!!

Ex 3)

Ex 4) → check using graphs or tables!

Ex 5)