## Calculus 8th Edition

Let $p(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} ... + a_{1}x + a_{0}$ where $a_{0}, a_{1}, a_{2}, ..., a_{n-1}, a_{n}$ are constants Using the limit laws, we have: $\\[1em]$ $\lim\limits_{x \to b} [p(x)] = \lim\limits_{x \to b} (a_{n}x^{n} + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} ... + a_{1}x^{1} + a_{0})$ $\\[1em]$ $= \lim\limits_{x \to b} (a_{n}x^{n}) + \lim\limits_{x \to b} (a_{n-1}x^{n-1}) + \lim\limits_{x \to b} (a_{n-2}x^{n-2}) + ... + \lim\limits_{x \to b} (a_{1}x) + \lim\limits_{x \to b} (a_{0})$ $\\[1em]$ $= \lim\limits_{x \to b} (a_{n}) \times \lim\limits_{x \to b} (x^{n}) + \lim\limits_{x \to b} (a_{n-1}) \times \lim\limits_{x \to b} (x^{n-1}) + \lim\limits_{x \to b} (a_{n-2}) \times \lim\limits_{x \to b} (x^{n-2}) + ... + \lim\limits_{x \to b} (a_{1}) \times\lim\limits_{x \to b} (x) + \lim\limits_{x \to b} (a_{0})$ $\\[1em]$ $= \lim\limits_{x \to b} (a_{n}) \times [\lim\limits_{x \to b} (x)]^{n} + \lim\limits_{x \to b} (a_{n-1}) \times [\lim\limits_{x \to b} (x)]^{n-1} + \lim\limits_{x \to b} (a_{n-2}) \times [\lim\limits_{x \to b} (x)]^{n-2} + ... + \lim\limits_{x \to b} (a_{1}) \times\lim\limits_{x \to b} (x) + \lim\limits_{x \to b} (a_{0})$ $\\[1em]$ $a_{n}\times b^{n} + a_{n-1}\times b^{n-1} + a_{n-2}\times b^{n-2} + ... + a_{1}\times b + a_{0} = p(b)$ $\\[1em]$ Thus $\lim\limits_{x \to b} [p(x)] = p(b)$