Base Case:
Let , and let be given. Take x such that .
Then, .
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.
Since a polynomial is built from constants, x, addition and multiplication, this statement is a corollary of two facts: , and . (Well, technically, you also need for a constant and .) 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 by induction on using the fact about the product of limits above.Let a be contained in C(complex numbers) and p is a polynomial, then p(x)-->p(a) as x-->a.
Whether the facts about the sum and product are taken for granted or need a proof depends on your course.