Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 53

Assume $P$ is a polynomial, show that $\lim \limits_{x \to a} P(x) = P(a)$

Let $P(x)$ be a polynomial function

$P(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$



$
\begin{equation}
\begin{aligned}

\lim \limits_{x \to a} P(x) =& a_0 + a_1 \lim \limits_{x \to a} x + a_2 \lim \limits_{x \to a} x^2 + ... + a_n \lim \limits_{x \to a} x^n\\
\\
\lim \limits_{x \to a} P(x) =& a_0 + a_1a + a_2a^2 + ... + a_n a^n\\
\\
P(a) =& a_0 + a_1(a) + a_2(a)^2 + ... + a_n(a)^n\\
\\
P(a) =& a_0 + a_1 a + a_2a^2 + ... + a_na^n\\
\\
.: \lim \limits_{x \to a} P(x) =& P(a)

\end{aligned}
\end{equation}
$