Proof using the Sum Rule

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Oct 17th 2010, 12:50 PM
zebra2147

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.
Oct 17th 2010, 01:02 PM
Math Major

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, $|P(a) - P(x)| = |c_1a + c_0 - (c_1x + c_0)| = |c_1(a - x)| = |c_1|*|a - x| < |c_1| * \frac{\epsilon}{|c_1|} = \epsilon$ .
Oct 17th 2010, 01:10 PM
zebra2147

Alright cool. That makes sense. Thanks.
Oct 17th 2010, 01:16 PM
emakarov

Quote:

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.