# L06 - Example Let f x = √ x 1 Evaluate 1 f(0 = 2 lim x...

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Lecture 6 — The Limit of a Function An Introduction Example: 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 3.001 3.01 3.1 f ( x ) 5.9 5.99 5.999 6.001 6.01 6.1 NOTE:

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To see what is happening graphically:
THE LIMIT OF A FUNCTION Deﬁnition: In our example, we write Example: For f ( x ) = x 2 - 1 , ﬁnd lim x 1 f ( x ) .

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We have the following picture: But consider Examples: f (3) = lim x 3 f ( x ) f ( - 1) = lim x →- 1 f ( x )
When the limit at a Point does not exist Example: Let f ( x ) = 1 x - 3 . What is lim x 3 f ( x ) ? We can try to ﬁnd the answer in two ways: 1) Consider the following table of values: x 2.9 2.99 2.999 3 3.001 3.01 3.1 f ( x ) -10 -100 -1000 1000 100 10 Find lim x 3 f ( x )

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2) We can also see this from the graph of f ( x ) = 1 x - 3 1) lim x 3 f ( x ) = 2) lim x 3 - f ( x ) = 3) lim x 3 + f ( x ) =
Example: Let f ( x ) = ± 2 x < 0 x 2 x 0 Its graph: Find: 1) f (0) 4) lim x 0 + f ( x ) 2) lim x 0 f ( x ) 5) lim x 2 f ( x ) 3) lim x 0 - f ( x )

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Example: 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)
TO EVALUATE LIMITS: Algebraic Methods:

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Unformatted text preview: Example: Let f ( x ) = √ x + 1 . Evaluate: 1) f (0) = 2) lim x → √ x + 1 NOTE: 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 . Properties of Limits Let L := lim x → a f ( x ) and M := lim x → a g ( x ) . Then for any real numbers c and r , 1) lim x → a c · f ( x ) = 2) lim x → a h f ( x ) ± g ( x ) i = 3) lim x → a h f ( x ) · g ( x ) i = 4) lim x → a f ( x ) g ( x ) = 5) lim x → a [ f ( x ) ] r = NOTE: If p ( x ) is a polynomial, then lim x → a p ( x ) = Example: lim x →-1 x 2 + 2 x-3 = Example: lim x →-3 x 2 + 3 x x-3 = Example: lim x → 3 x 2-9 x-3 =...
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L06 - Example Let f x = √ x 1 Evaluate 1 f(0 = 2 lim x...

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