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lecture6 - Lecture 6 10 Some Basic Limits 3 lim x → a b =...

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Lecture 6 1 LECTURE 6, Sec. 2.4 THE LIMIT OF A FUNCTION An Introduction ex. Let f ( x ) = x 2 - 9 x - 3 Consider the following table of values for our func- tion: x 2.9 2.99 2.999 3 3.01 3.001 y 5.9 5.99 5.999 6.01 6.001 NOTE:
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Lecture 6 2 To see what is happening graphically: 6 - ?
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Lecture 6 3 THE LIMIT OF A FUNCTION Def. In our example, we write ex. For f ( x ) = x 2 - 1, find lim x 1 f ( x ).
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Lecture 6 4 We have the following picture: But consider ex. 6 - ? f (3) = f ( - 1) = lim x 3 f ( x ) = lim x →- 1 f ( x ) =
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Lecture 6 5 When the Limit at a Point does not Exist ex. Let f ( x ) = 1 x - 3 . What is lim x 3 f ( x )? We can try to find the answer two ways: 1) Consider the following table of values: x 2.9 2.99 2.999 3.001 3.01 3.1 y -10 -100 -1000 1000 100 10 Find lim x 3 f ( x )
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Lecture 6 6 2) We can also see this from the graph of f ( x ) = 1 x - 3 6 - ? 1) lim x 3 f ( x ) = 2) lim x 3 - f ( x ) = 3) lim x 3 + f ( x ) =
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Lecture 6 7 ex. f ( x ) = 2 if x < 0 x 2 if x 0 Its graph: 6 - ? Find: f (0) lim x 0 + f ( x ) lim x 0 f ( x ) lim x 2 f ( x ) lim x 0 - f ( x )
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Lecture 6 8 ex. 6 - ? Evaluate: 1) lim x →- 3 f ( x ) 5) lim x 2 f ( x ) 2) f (0) 6) lim x 2 - f ( x ) 3) lim x 0 f ( x ) 7) lim x 2 + f ( x ) 4) f (2)
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Lecture 6 9 TO EVALUATE LIMITS: ALGEBRAIC METHODS ex. Let f ( x ) = x + 1. Evaluate: 1) f (0) = 2) lim x 0 x
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Unformatted text preview: Lecture 6 10 Some Basic Limits 3) lim x → a b = 4) lim x → a x = 5) lim x → a x n = 6) lim x → a n √ x = for appropriate values of x and a . Lecture 6 11 Properties of Limits (p. 102) Let lim x → a f ( x ) = L , and let lim x → a g ( x ) = M . Then for any real numbers c and r , 1) lim x → a cf ( x ) = c [lim x → a f ( x )] = 2) lim x → a [ f ( x ) ± g ( x )] = lim x → a f ( x ) ± lim x → a g ( x ) = 3) lim x → a [ f ( x ) g ( x )] = [lim x → a f ( x )][lim x → a g ( x )] = 4) lim x → a f ( x ) g ( x ) = lim x → a f ( x ) lim x → a g ( x ) = 5) lim x → a [ f ( x )] r = [lim x → a f ( x )] r = Lecture 6 12 NOTE: If p ( x ) is a polynomial, then lim x → a p ( x ) = ex. lim x →-1 x 2 + 2 x-3 = ex. lim x →-3 x 2 + 3 x x-3 = ex. lim x → 3 x 2-9 x-3 =...
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