# Thread: Proof using the Sum Rule

1. ## 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.

2. Base Case:

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

Then, $\displaystyle |P(a) - P(x)| = |c_1a + c_0 - (c_1x + c_0)| = |c_1(a - x)| = |c_1|*|a - x| < |c_1| * \frac{\epsilon}{|c_1|} = \epsilon$.

3. Alright cool. That makes sense. Thanks.

4. 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: $\displaystyle \lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x) + \lim_{x\to a}g(x)$, and $\displaystyle \lim_{x\to a}(f(x)\cdot g(x))=(\lim_{x\to a}f(x))\cdot (\lim_{x\to a}g(x))$. (Well, technically, you also need $\displaystyle \lim_{x\to a}c=c$ for a constant $\displaystyle c$ and $\displaystyle \lim_{x\to a}x=a$.) 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 $\displaystyle \lim_{x\to a} x^n=a^n$ by induction on $\displaystyle n$ 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.