Section 2.3 Part Two - itself and if lim lim x c x c h x L...

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Chapter Two: Limits Section 2.3 con’d Theorem 2.7 (Limit of function that agrees with ( ) f x at all points but one) Let c be a real number and let ( ) ( ) f x g x = for all x c in an open interval containing c. If the limit of ( ) g x as x approaches c exists, then the limit of ( ) f x also exists and ( ) ( ) lim lim x c x c f x g x = . Whenever you are algebraically solving a limit, as soon as you cancel, you are using this theorem, and the resultant function is the ( ) g x . When you apply direct substitution, if you get the answer 0 0 , this is called an indeterminate form. 0 0 is never, ever your final answer. It simply means that direct substitution fails. It does not tell you anything about the limit. Squeeze Theorem : If ( ) ( ) ( ) h x f x g x for all x in an open interval containing c, except possibly at c
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Unformatted text preview: itself, and if ( ) ( ) lim lim x c x c h x L g x → → = = then ( ) lim x c f x → exists and is equal to L. • Three Special Limits 1. sin lim 1 x x x → = 2. 1 cos lim x x x →-= 3. ( ) 1 lim 1 x x x e → + = Strategy for Finding Limits: 1. Try direct substitution: One of three things will happen: a. If you get a real number, then you are done. b. If you get , then you have to algebraically manipulate your function so that you can use direct substitution: i. Factoring ii. Rationalizing (Radicals) iii. Simplifying Complex Rational Expressions iv. Special Limits c. Will be seen later d. If none of the top three apply, try using the Squeeze Theorem....
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This note was uploaded on 06/21/2011 for the course MTH 150 taught by Professor Marioborha during the Summer '11 term at Moraine Valley Community College.

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Section 2.3 Part Two - itself and if lim lim x c x c h x L...

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