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

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


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