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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
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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
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 : • Direct substitution • Simplify to an identical function except at x=c ; then use direct substitution. • Use a graph or table to check. • 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 4) → check using graphs or tables!