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Unformatted text preview: + m = x . or the same 1 reason, lim + M = x . [Remember that were still only considering positive so we can only talk about right limits.] 10. By the continuity of f , this means that lim + f ( m ) = lim + f ( M ) = f ( x ). 11. Taking limits preserves nonstrict inequalities and we want to take the limit of the expression f ( m ) h ( ) f ( M ) from Step 8 as + . 12. From Step 10 , we know that f ( m ) f ( x ) and f ( M ) f ( x ). By the Squeeze Theorem , this means that lim + h ( ) = f ( x ). 13. We can similarly perform Steps 4-12 for the case where < 0 and we get lim -h ( ) = f ( x ). 14. The previous two steps imply that lim h ( ) = f ( x ). 15. Since g ( x ) = lim h ( ) by the way we have dened h , the previous step shows that g ( x ) = f ( x ). 2...
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This note was uploaded on 06/26/2010 for the course MATH 1A taught by Professor Wilkening during the Fall '08 term at University of California, Berkeley.
- Fall '08