Enter the rule for $f\left(x\right)$ in the space above the "Asymptotes" button. Press that button to have the equations of the asymptotes appear in the spaces to the right. There are no horizontal asymptotes, but two vertical asymptotes (the lines $x\=-2$, and $x\=3$). In addition, the line $y\=2 x-1$ is an oblique asymptote because the degree of the numerator is one more than the degree of the denominator.

Press the "Rational Function Tutor" button to launch the Rational Function tutor. Figure 1.5.6(a) is a screenshot of this tutor as it appears after it is launched.

If the tutor is launched from the Tools≻Tutors: Precalculus menu, the numerator and denominator of the rational function $f$ have to be entered separately. By launching it from the task template, this data-entry chore is simplified. In the space below the numerator and denominator, the equations of the asymptotes appear. However, when this tutor is closed, all this information is lost, and only the graph is preserved.

Accessing this tutor from the task template preserves the equations of the asymptotes, and also the graph, which is written to the task template when the Close button on the tutor is pressed. The Maple command (RationalFunctionPlot) at the bottom of the tutor can be copied, then pasted and executed; its output is the graph seen in the tutor.

Tools≻Load Package: Student Calculus 1

Control-drag (or type) $f\left(x\right)\=\dots$ 

Context Panel: Assign Function

Apply the Asymptotes command.

Expression palette: Limit operator

Expression palette: Limit operator

Expression palette: Limit operator

Expression palette: Limit operator

Expression palette: Limit operator

Expression palette: Limit operator

Type $f\left(x\right)$ and press the Enter key.

Context Panel: Numerator

Context Panel: Assign to a Name≻$p$

Type $f\left(x\right)$ and press the Enter key.

Context Panel: Numerator

Context Panel: Assign to a Name≻$q$

Implement long division of $p$ by $q$ via the quo command. The remainder is assigned to $r$.

Divide the numerator by $x^2$.

Divide the denominator by $x^2$.

Control-drag to form $\frac{p\/x^2}{q\/x^2}$.
The dominant term in the numerator is $2 x$, from which the limits at $\pm \infty$ readily follow.

Obtain the zeros of the denominator.

Context Panel: Evaluate at a Point≻$x\=-2$

Context Panel: Evaluate at a Point≻$x\=3$ 

See Figure 1.5.6(b) for a graph of the denominator.

To the left of $x\=-2$, the denominator is positive and the numerator is negative (approximately $-45$).
Consequently, $f\to -\infty$ as $x\to -2$ from the left.

To the right of $x\=-2$, the denominator is negative and the numerator is negative (approximately $-45$).
Consequently, $f\to \+\infty$ as $x\to -2$ from the right.

To the left of $x\=3$, the denominator is negative and the numerator is positive (approximately 35).
Consequently, $f\to -\infty$ as $x\to 3$ from the left.

To the right of $x\=3$, the denominator is positive and the numerator is positive (approximately 35).
Consequently, $f\to \+\infty$ as $x\to 3$ from the right.