exercises_9 - 3 x 2 + 3 x 2-3 ; (b) lim x 1 x 2-2 x + 1 x...

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Analysis 1: Exercises 9 1. (a) Prove that lim x a f ( x ) = lim h 0 f ( a + h ) . (b) Prove that lim x a f ( x ) = b if and only if lim x a ± f ( x ) - b ² = 0. 2. Prove the following two theorems. Theorem 1. Let lim x a f ( x ) = A and lim x a g ( x ) = B . Then (i) lim x a ( f ( x ) + g ( x ) ) = A + B ; (ii) lim x a ( f ( x ) · g ( x ) ) = A · B . If in addition B 6 = 0 , then (iii) lim x a ³ f ( x ) g ( x ) ´ = A B . Theorem 2. Let lim x a f ( x ) = A and lim x a g ( x ) = B . Suppose that [( δ > 0) h { x R | 0 < | x - a | < δ } ⊆ D ( f ) D ( g ) i and ( 0 < | x - a | < δ ) ± f ( x ) g ( x ) ² . Then A B ; that is, lim x a f ( x ) lim x a g ( x ) . 3. Let lim x 0 f ( x ) = 0. Prove that (a) lim x 0 f (2 x ) = 0; (b) lim x 0 f ( x 2 ) = 0. 4. Compute the following limits: (a) lim x
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Unformatted text preview: 3 x 2 + 3 x 2-3 ; (b) lim x 1 x 2-2 x + 1 x 3-x ; (c) lim x 1 x-1 x-1 . 5. We dene lim x a f ( x ) = + to mean that ( M R )( &gt; 0)( x D ( f )) h ( &lt; | x-a | &lt; ) ( f ( x ) &gt; M ) i . (a) Prove that lim x 3 1 ( x-3) 2 = + . 2 (b) Dene lim x a + f ( x ) = + and prove that lim x 0+ 1 x = + ....
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exercises_9 - 3 x 2 + 3 x 2-3 ; (b) lim x 1 x 2-2 x + 1 x...

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