Lecture1 - 4 Constant Multiple Rule The limit of a constant times a function is the constant times the limit of the function 5 Quotient Rule The

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2.2 Limit of a Function and Limit Laws Limits of Function Values Example 1 How does the function f ( x ) = x 2 - 1 x - 1 behave near x = 1? 1
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Let f ( x ) be defined on an open interval about x 0 , except possibly at x 0 itself . If f ( x ) gets arbitrarily close to L (as close to L as we like) for all x sufficiently close to x 0 , we say that which is read “the limit of f ( x ) as x approaches x 0 is L .” lim x x 0 x = lim x x 0 k = any constant k Example For the function f ( t ) graphed here, find the following limits or explain, why they do not exist. a) lim t →- 2 f ( t ) b) lim t →- 1 f ( t ) c) lim t 0 f ( t ) 2
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Example 4 Discuss the behavior of the following functions as x 0. a) U ( x ) = ± 0 , x < 0 1 , x 0 3
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b) g ( x ) = ± 1 x , x 6 = 0 0 , x = 0 c) f ( x ) = ± 0 , x 0 sin 1 x , x > 0 4
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The Limit Laws THEOREM 1 - Limit Laws If L,M,c and k are real numbers and lim x c f ( x ) = L and lim x c g ( x ) = M , then 1. Sum Rule: The limit of the sum of two functions is the sum of their limits. 2. Difference Rule: The limit of the difference of two functions is the difference of their limits. 3. Product Rule: The limit of a product of two functions is the product of their limits.
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Unformatted text preview: 4. Constant Multiple Rule: The limit of a constant times a function is the constant times the limit of the function. 5. Quotient Rule: The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero. 6. Power Rule: The limit of a rational power of a function is that power of the limit of the function, provided the limit power is a real number. 5 Example 5 lim x → c x 4 + x 2-1 x 2 +5 = lim x →-2 √ 4 x 2-3 = THEOREM 2 - Limits of Polynomials If P ( x ) = a n x n + a n-1 x n-1 + ... + a , then THEOREM 3 - Limits of Rational Functions If P ( x ) and Q ( x ) are polynomials and Q ( c ) 6 = 0, then 6 Example lim x →-7 (2 x + 5) Example lim x → 5 4 x-7 Example 6 lim x →-1 x 3 +4 x 2-3 x 2 +5 = 7...
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This note was uploaded on 03/31/2008 for the course MATH 1205 taught by Professor Fbhinkelmann during the Spring '08 term at Virginia Tech.

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Lecture1 - 4 Constant Multiple Rule The limit of a constant times a function is the constant times the limit of the function 5 Quotient Rule The

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