lecture2_1

lecture2_1 - 1-cos x ≤ | x | and the Sanwich Theorem to...

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2.2 Limit of a Function and Limit Laws Eliminating Zero Denominators Algebraically Example 7 Factoring lim x 1 x 2 + x - 2 x 2 - x = Example 9 Mutliplying by Conjugate lim x 0 x 2 +100 - 10 x 2 = 1

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The Sandwish Theorem THEOREM 4 - The Sandwich Theorem Suppose that for all x in some open interval containing c , except possibly at x = c itself. Suppose also that Then The Sanwich Theorem is also called the Squeeze Theorem or the Pinching Theorem . Example 10 Given that 1 - x 2 4 u ( x ) 1 + x 2 2 for all x 6 = 0 , ﬁnd lim x 0 u ( x ), no matter how complicated u is. 2
Example 11 a) Use the fact that -| x | ≤ sin x ≤ | x | and the Sanwich Theorem to show that lim x 0 sin x = 0. b) Use the fact that 0

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Unformatted text preview: 1-cos x ≤ | x | and the Sanwich Theorem to show that lim x → cos x = 1. Hint: What is lim x → (1-cos x )? 3 c) For any function f ( x ), if lim x → c | f ( x ) | = 0, then lim x → f ( x ) = 0. THEOREM 5 If for all x in some open interval containing c , except possibly at x = c itself, and the limits of f and g both exist as x approaches c , then Example 12 If lim x →-2 f ( x ) x 2 = 1, ﬁnd a) lim x →-2 f ( x ) 4 b) lim x →-2 f ( x ) x Example 14 lim x → sin 2 x lim x → √ 1 + cos 2 x 5...
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lecture2_1 - 1-cos x ≤ | x | and the Sanwich Theorem to...

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