Saturday, December 23, 2017

Calculus and Its Applications, Chapter 1, 1.2, Section 1.2, Problem 16

Determine the lim 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 -1} \left( 3x^5 + 4x^4 - 3x + 6 \right) &= \lim_{x \to -1} 3x^5 + \lim_{x \to -1} 4x^4 - \lim_{x \to -1} 3x + \lim_{x \to -1} 6 && \text{The limit of a difference is the difference of the limits and the limit of a sum is the sum of the limits}\\ \\ &= 3 \cdot \lim_{x \to -1} x^5 + 4 \cdot \lim_{x \to -1} x^4 - 3 \cdot \lim_{x \to -1}x + \lim_{x \to -1} 6 && \text{The limit of a constant times a function is the constant times the limit}\\ \\ &= 3 \cdot \left( \lim_{x \to -1}x \right)^5 + 4 \cdot \left( \lim_{x \to -1}x \right)^4 - 3 \cdot \lim_{x \to -1} x + \lim_{x \to -1} 6 && \text{The limit of a power is the power of the limit}\\ \\ &= 3 \cdot \left( \lim_{x \to -1}x \right)^5 + 4 \cdot \left( \lim_{x \to -1}x \right)^4 - 3 \cdot \lim_{x \to -1}x + 6 && \text{The limit of a constant is the constant}\\ \\ &= 3(-1)^5 + 4(-1)^4 - 3(-1) + 6 && \text{Substitute } -1\\ \\ &= -3 + 4 +3 + 6\\ \\ &= 10 \end{aligned} \end{equation}

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