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Limit of Functions

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We prove two results involving the limit of functions.

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1.2.20) First consider the case n=0 (base case). In this case we have f(x) = c_0, whence lim_{x->a}f(x) = c_0 = f(a). This proves the base case. Now for the induction step. We assume lim_{x->a}g(x) = f(0) for all polynomials g(x) of fixed degree n-1. Now let f(x) be a polynomial of degree n. We may write f(x) = c_n x^n + g(x), where g(x) is a polynomial of degree n-1. We have

lim_{x->a}f(x) = c_n lim_{x->a} x^n + lim_{x->a} g(x)
= ...

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