a3 242 - MATH 242 Analysis 1(Fall 2011 Assignment 3 Due in...

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Unformatted text preview: MATH 242 Analysis 1 (Fall, 2011) Assignment 3. Due in class Tuesday, October 18. Instructions: This assignment is out of 50 points. You may choose any questions whose points add up to 50. As in the previous assignment, provide all details in your work. Also as previously, stars (*) indicate possibly more challenging questions. 1. (10 pts.) Using the ε- δ definition of a limit, show that (a) lim x → c x 3 = c 3 for any c ∈ R ; (b) lim x → 1 x 2- x +1 x +1 = 1 2 . 2. (10 pts.) Let f and g be real-valued functions defined on A ⊆ R and let c ∈ R be a cluster point of A . Suppose that f is bounded on a neighborhood of c and that lim x → c g ( x ) = 0. Prove that lim x → c f ( x ) g ( x ) = 0. Remark: Recall that a neighborhood of a point x is defined as V δ ( x ) = { x ∈ R : | x- x | < δ } = ( x- δ, x + δ ). In the context of problem 2, the expression ” f is bounded on a neighborhood of c ” means that, for some δ > 0, function f ( x ) for x in A ∩ V δ ( x ) is bounded.) is bounded....
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a3 242 - MATH 242 Analysis 1(Fall 2011 Assignment 3 Due in...

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