4130sols7 - 4130 HOMEWORK 7 Due Tuesday April 13 (1) Let D...

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Unformatted text preview: 4130 HOMEWORK 7 Due Tuesday April 13 (1) Let D R . Let f,g : D R and let a be a cluster point of D . Suppose lim x a f ( x ) = L and lim x a g ( x ) = M . Show that lim x a f ( x ) g ( x ) = LM . First, we show that f ( x ) and g ( x ) are bounded near a . Since lim x a f ( x ) = L , there exists > 0 such that if x D and | x- a | < then | f ( x )- L | < 1, whence | f ( x ) | | f ( x )- L | + | L | = 1 + | L | . Similarly, there exists such that if | x- a | < then | g ( x ) | 1+ | M | . Replacing by the minimum of and , we may assume that | x- a | < implies | f ( x ) | 1 + | L | , | g ( x ) | 1 + | M | . Now, there is 1 > 0 such that | x- a | < 1 implies | f ( x )- L | < / 2(1 + | M | ). Similarly, there is 2 > 0 such that | x- a | < 2 implies | g ( x )- M | < / 2(1 + | L | ). Let 3 = min { 1 , 2 , } . Then | x- a | < 3 implies: | f ( x ) g ( x )- LM | | f ( x ) || g ( x )- M | + | M || f...
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4130sols7 - 4130 HOMEWORK 7 Due Tuesday April 13 (1) Let D...

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