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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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