Calculus and Its Applications, Chapter 1, 1.2, Section 1.2, Problem 14

Determine the $\displaystyle \lim_{x \to 5} (x^2 - 6x + 9)$ by using the Theorem on Limits of Rational Functions.
When necessary, state that the limit does not exist.


$\begin{equation} \begin{aligned} \lim_{x \to 5} (x^2 - 6x + 9) &= \lim_{x \to 5} x^2 - \lim_{x \to 5} 6x + \lim_{x \to 5} 9 && \text{The limit of a difference is the difference of the limits and the limit of a sum is the sum of the limits}\\ \\ &= \left( \lim_{x \to 5} x\right)^2 - \lim_{x \to 5} 6x + \lim_{x \to 5} 9 && \text{The limit of a power is the power of the limit}\\ \\ &= \left( \lim_{x \to 5}x \right)^2 - 6 \cdot \lim_{x \to 5} x + \lim_{x \to 5} 9 && \text{The limit of a constant times a function is the constant times the limit}\\ \\ &= \left( \lim_{x \to 5}x \right)^2 - 6 \cdot \lim_{x \to 5} x + 9 && \text{The limit of a constant is the constant}\\ \\ &= (2)^2 - 6 \cdot 2 + 9 && \text{Substitute 2}\\ \\ &= 4 - 12 + 9\\ \\ &= 1 \end{aligned} \end{equation}$