# The Limit Polynomial Rule $ \lim_{x \to a} p(x) = p(x) $ ^2f957s ## Proof We can write any polynomial as $ p(x) = \sum_{k=0}^n c_kx^k $ By combining the constant, sum and power rules proved above we get: $ \begin{align} \lim_{x \to a} p(x) &= \lim_{x \to a} \left[ \sum_{k=0}^n c_kx^k \right] \\ &= \sum_{k=0}^n \lim_{x \to a} \left[ c_kx^k \right] \\ &= \sum_{k=0}^n c_k \lim_{x \to a} x^k \\ &= \sum_{k=0}^n c_k a^k \\ &= p(a) \end{align} $