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# 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 δ > 0 such that if x ∈ D and | x- a | < δ then | f ( x )- L | < 1, whence | f ( x ) | ≤ | f ( x )- L | + | L | = 1 + | L | . Similarly, there exists δ such that if | x- a | < δ then | g ( x ) | ≤ 1+ | M | . Replacing δ by the minimum of δ and δ , we may assume that | x- a | < δ implies | f ( x ) | ≤ 1 + | L | , | g ( x ) | ≤ 1 + | M | . Now, there is δ 1 > 0 such that | x- a | < δ 1 implies | f ( x )- L | < ε/ 2(1 + | M | ). Similarly, there is δ 2 > 0 such that | x- a | < δ 2 implies | g ( x )- M | < ε/ 2(1 + | L | ). Let δ 3 = min { δ 1 ,δ 2 ,δ } . Then | x- a | < δ 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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