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Math Help - Proof using the Sum Rule

  1. #1
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    Proof using the Sum Rule

    I'm studying for a test and I came across this exercise...
    Let a be contained in C(complex numbers) and p is a polynomial, then p(x)-->p(a) as x-->a. The hint given says to use the Sum Rule and induction on the degree but I still am not sure how to get started. Any guidance would be appreciated.
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  2. #2
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    Base Case:

    Let  P(x) = c_1x + c_0 , and let  \epsilon > 0 be given. Take x such that  |x - a| < \frac{\epsilon}{|c_1|} .

    Then,  |P(a) - P(x)| = |c_1a + c_0 - (c_1x + c_0)| = |c_1(a - x)| = |c_1|*|a - x| < |c_1| * \frac{\epsilon}{|c_1|} = \epsilon .
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  3. #3
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    Alright cool. That makes sense. Thanks.
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  4. #4
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    Let a be contained in C(complex numbers) and p is a polynomial, then p(x)-->p(a) as x-->a.
    Since a polynomial is built from constants, x, addition and multiplication, this statement is a corollary of two facts: \lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x) + \lim_{x\to a}g(x), and \lim_{x\to a}(f(x)\cdot g(x))=(\lim_{x\to a}f(x))\cdot (\lim_{x\to a}g(x)). (Well, technically, you also need \lim_{x\to a}c=c for a constant c and \lim_{x\to a}x=a.) It is suggested that you break a polynomial into a sum of monomials and use the fact about the sum of limits above. Also, it is suggested that you show that \lim_{x\to a} x^n=a^n by induction on n using the fact about the product of limits above.

    Whether the facts about the sum and product are taken for granted or need a proof depends on your course.
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