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Lesson 3 Limits (Def and Theorems)

# Lesson 3 Limits (Def and Theorems) - ma b Theorem 4 Limit...

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LIMITS OF FUNCTION

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At the end of this lesson, the students are expected to perform the following: define limits illustrate limits and its theorems evaluate limits applying the given theorems
Consider the function Note that the given function is not defined at x = 1 since at this point f(x) has the form 0/0 which is meaningless or indeterminate. But what will happen to f(x) as x approaches 1, or is f(x) approaching some specific values as x approaches 1? - = - 3 1 ( ) 1 x f x x

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- = - 3 1 ( ) 1 x f x x Consider the function: x 1.2 1.1 1.0 1 1 0.9 9 0.9 0.8 0.7 3.6 4 3.3 1 3.0 3 ? 2.9 7 2.7 1 2.3 1 2.1 9 3 1 1 x y x - = -
From the illustration, we arrived at the conclusion that “f(x) approaches 3 as x approaches 1” . In symbol we write: 3 1 1 lim 3 1 x x x - = - read as “the limit of as x approaches 1 is 3.” x 3 - 1 x - 1

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Theorem 1: Limit of a Constant If c is a constant, then for any number a lim x ® a C = C
Theorem 2: Limit of the Identify Function a x a lim x =

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Theorem 3: Limit of a Linear Function If m and b are constants lim x ® a ( mx

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Unformatted text preview: ma + b Theorem 4: Limit of the Sum or Difference of Functions lim x ® a f ( x ) ± g ( x ) [ ] = lim x ® a f ( x ) ± lim x ® a g ( x ) = L ± M If lim x® a f ( x ) = L and lim x® a g(x) = M,then Theorem 5: Limit of the Product lim x ® a f ( x )· g ( x ) [ ] = lim x ® a f ( x )· lim x ® a g ( x ) = L · M If lim x® a f(x) = L and lim x® a g(x) = M ,then Theorem 6: Limit of the n th Power of a function lim x ® a f ( x ) [ ] n = L n then integer, positive any is n and L f(x) lim If a x → = Theorem 7: Limit of a Quotient lim x ® a f ( x ) g ( x ) = lim x ® a f ( x ) lim x ® a g ( x ) = L M , M ¹ 0 If lim x® a f ( x ) = L and lim x ® a g ( x ) = M , then Theorem 8: Limit of the n th Root of a Function lim x ® a f ( x ) n = lim x ® a f ( x ) n = L n then , L f(x) lim and integer positive a is n If a x → = Note: If n is even then L ≥...
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