Activity 5.4.3.
Let \(s(x) = \frac{3x - 5}{7x^2 + 2x - 11}\) and \(u(x) = \frac{3x^2 - 5x + 1}{7x + 2}\text{.}\) Note that both the numerator and denominator of each of these rational functions increases without bound as \(x \to \infty\text{,}\) and in addition that \(x^2\) is the highest order term present in each of \(s\) and \(u\text{.}\)
(a)
Using a similar algebraic approach to our work in Activity 5.4.2, multiply \(s(x)\) by \(1 = \frac{\frac{1}{x^2}}{\frac{1}{x^2}}\) and hence evaluate
\begin{equation*}
\lim_{x \to \infty} \frac{3x - 5}{7x^2 + 2x - 11}\text{.}
\end{equation*}
What value do you find?
(b)
Plot the function \(y = s(x)\) on the interval \([-10,10]\text{.}\) What is the graphical meaning of the limit you found in (a)?
(c)
Next, use appropriate algebraic work to consider \(u(x)\) and evaluate
\begin{equation*}
\lim_{x \to \infty} \frac{3x^2 - 5x + 1}{7x + 2}\text{.}
\end{equation*}
What do you find?
(d)
Plot the function \(y = u(x)\) on the interval \([-10,10]\text{.}\) What is the graphical meaning of the limit you computed in (c)?
