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lecture12

# lecture12 - Math 006(Lecture 12 Introduction to Limits...

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Math 006 (Lecture 12) Introduction to Limits Example 1. Graph the function f ( x ) = x 2 - 1 x + 1 . Definition 1. We write lim x c f ( x ) = L or f ( x ) L as x c if the function value f ( x ) is close to the single real number L whenever x is close to, but not equal to c . Example 2. Let h ( x ) = | x | x . Find h (0) and lim x 0 h ( x ). We saw that the values of the function h ( x ) approached two different numbers, depending on the direction of approach, and it was natural to refer to these value as “the limit from the left” and “the limit from the right”. Definition 2. We write lim x c - f ( x ) = K, ( lim x c + f ( x ) = H ) and call K the left-hand limit of f ( x ) ( H the right-hand limit of f ( x )) whenever x is close to c , but to the left (right) of c . 1

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Example 3. Find lim x 0 + h ( x ) and lim x 0 - h ( x ). Theorem 1. A limit exists if and only if the corresponding left-hand limit and the right- hand limit are equal . That is, lim x c f ( x ) = L if and only if lim x c - f ( x ) = L and lim x c + f ( x ) = L Example 4. Find the function value of f at c , f ( c
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