Single Variable Calculus, Chapter 4, 4.4, Section 4.4, Problem 12

Determine $\displaystyle \lim_{x \to \infty} \frac{2 - 3y^2}{5y^2 + 4y}$


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\begin{equation}
\begin{aligned}

\lim_{x \to \infty} \frac{2 - 3y^2}{5y^2 + 4y} \cdot \frac{\displaystyle \frac{1}{y^2}}{\displaystyle\frac{1}{y^2}}
=& \lim_{x \to \infty} \frac{\displaystyle \frac{2}{y^2} - \frac{3 \cancel{y^2} }{\cancel{y^2}}}{\displaystyle \frac{5\cancel{y^2}}{\cancel{y^2}} + \frac{4y}{y^2}}
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=& \lim_{x \to \infty} \frac{\displaystyle \frac{2}{y^2} - 3}{\displaystyle 5 + \frac{4}{y}}
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=& \frac{\displaystyle \lim_{x \to \infty} \left( \frac{2}{y^2} - 3 \right) }{ \lim_{x \to \infty} \left( 5 + \frac{4}{y} \right)}
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=& \frac{\displaystyle \lim_{x \to \infty} \frac{2}{y^2} - 3}{\displaystyle 5 + \lim_{x \to \infty} \frac{4}{y}}
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=& \frac{0 - 3}{5 + 0}
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=& \frac{-3}{5}

\end{aligned}
\end{equation}
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