Activity 5.4.2.
Consider the rational function \(r(x) = \frac{3x^2 - 5x + 1}{7x^2 + 2x - 11}\text{.}\)
Observe that the largest power of \(x\) that’s present in \(r(x)\) is \(x^2\text{.}\) In addition, because of the dominant terms of \(3x^2\) in the numerator and \(7x^2\) in the denominator, both the numerator and denominator of \(r\) increase without bound as \(x\) increases without bound. In order to understand the long-range behavior of \(r\text{,}\) we choose to write the function in a different algebraic form.
(a)
Note that we can multiply the formula for \(r\) by the form of \(1\) given by \(1 = \frac{\frac{1}{x^2}}{\frac{1}{x^2}}\text{.}\) Do so, and distribute and simplify as much as possible in both the numerator and denominator to write \(r\) in a different algebraic form.
(b)
Having rewritten \(r\text{,}\) we are in a better position to evaluate \(\lim_{x \to \infty} r(x)\text{.}\) Using our work from (a), we have
\begin{equation*}
\lim_{x \to \infty} r(x) = \lim_{x \to \infty} \frac{3 - \frac{5}{x} + \frac{1}{x^2}}{7 + \frac{2}{x} - \frac{11}{x^2}}\text{.}
\end{equation*}
What is the exact value of this limit and why?
(c)
Next, determine
\begin{equation*}
\lim_{x \to -\infty} r(x) = \lim_{x \to -\infty} \frac{3 - \frac{5}{x} + \frac{1}{x^2}}{7 + \frac{2}{x} - \frac{11}{x^2}}\text{.}
\end{equation*}
(d)
Use Desmos to plot \(r\) on the interval \([-10,10]\text{.}\) In addition, plot the horizontal line \(y = \frac{3}{7}\text{.}\) What is the meaning of the limits you found in (b) and (c)?
