Analyzing Limits in Calculus
Concept of evaluating limits analytically in calculus, focusing on properties of limits, direct substitution, special trigonometric limits, and techniques like dividing out and rationalizing. Includes examples and explanations to enhance understanding.
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Presentation Transcript
Chapter Chapter 1 1.3: Evaluating Limits Analytically Evaluating Limits Analytically .3: HONORS CALCULUS/CALCULUS HONORS CALCULUS/CALCULUS
Properties of Limits The limit of f(x) as x approaches c does not depend on the value of f at x = c. However, the limit could be precisely f(c). In such cases, the limit can be evaluated by direct substitution. That is, ??? ? ?? ? = ?(?) Such well-behaved functions are continuous at c.
Ex. 1) ? ???= ??? ? ?? = ??? ? ?? = ???
Ex. 2) ? ?(???+ ?) = ???
Ex. 3) ??+ ? + ? ? + ? ??? ? ? =
Ex. 4) ? ?????? = ??? ? ????? = ??? ??? ? ?(?????) =
Ex. 5) ?? ? ? ?= ??? ? ?
Ex. 5) ?? ? ? ?= ??? ? ?
DIVIDING OUT TECHNIQUE Ex. 6) ??+ ? ? ? + ? ??? ? ? =
RATIONALIZING TECHNIQUE Ex. 7) ? + ? ? ? ??? ? ? =
RATIONALIZING TECHNIQUE Ex. 8) ?/(? + ?) ?/? ? ??? ? ?
THE SQUEEZE THEOREM THEOREM 1.9: TWO SPECIAL TRIGONOMETRIC LIMITS ? ???? ? ???? ? ??? ? ? = ? ??? ? ? = ?
Ex. 9) ????? ?? ??? ? ? =
Ex. 10) ????? ?? ??? ? ? =
Ex. 11) ? ????? ?? ??? ? ? =
Ex. 12) ???? ? ??? ? ? =
Ex. 13) ????? ?? ??? ? ? =
