Single Variable Calculus, Chapter 4, 4.4, Section 4.4, Problem 10

Determine $\displaystyle \lim_{x \to \infty} \frac{3x + 5}{x - 4}$


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\begin{equation}
\begin{aligned}

\lim_{x \to \infty} \frac{3x + 5}{x - 4} \cdot \frac{\displaystyle \frac{1}{x}}{\displaystyle \frac{1}{x}} =& \frac{\displaystyle \frac{3 \cancel{x}}{\cancel{x}} + \frac{5}{x} }{\displaystyle \frac{\cancel{x}}{\cancel{x}} - \frac{4}{x}}
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=& \lim_{x \to \infty} \frac{\displaystyle 3 + \frac{5}{x}}{\displaystyle 1 - \frac{4}{x}}
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=& \frac{\displaystyle \lim_{x \to \infty} \left( 3 + \frac{5}{x} \right) }{\displaystyle \left( 1 - \frac{4}{x} \right) }
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=& \frac{3 + \displaystyle \lim_{x \to \infty} \frac{5}{x}}{\displaystyle 1 - \displaystyle \lim_{x \to \infty} \frac{4}{x}}
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=& \frac{3 + 0}{1 - 0}
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=& \frac{3}{1}
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=& 3


\end{aligned}
\end{equation}
$