Chapter 1: Limits
Section 1.5: Limits at Infinity and Infinite Limits
Example 1.5.1
Evaluate limx→∞pxqx, where p and q are respectively, the cubic polynomials 4 x3+5 x2+6 x+7, and 7 x3+6 x2+5 x+4.
Solution
Enter the data
Control-drag (or type) p.
Context Panel: Assign to a Name≻p
4 x3+5 x2+6 x+7→assign to a namep
Control-drag (or type) q.
Context Panel: Assign to a Name≻q
7 x3+6 x2+5 x+4→assign to a nameq
Apply Maple's limit operator
Expression palette: Limit operator
Context Panel: Evaluate and Display Inline
limx→∞pq = 47
Draw a graph
Code for Figure 1.5.1(a) is hidden in the cell containing the graph.
To obtain Figure 1.5.1(a) interactively, write the sequence pq,47 and invoke the Plot Builder from the Context Panel.
The line y=4/7 is a horizontal asymptote.
(The relevant options for the Plot Builder are: Click Edit in the context panel beside each plot thumbnail to set the range for x and change the color of the curve, and then in Global Options set the view for axis[2].)
use plots in module() local p1,p2,p3,p,q; p:=4*x^3+5*x^2+6*x+7; q:=7*x^3+6*x^2+5*x+4; p1:=plot([4/7,p/q],x=0..20,y=0..2,color=[red,black]): p2:=textplot([10,.4,typeset(y=4/7)]): p3:=display(p1,p2); print(p3); end module: end use:
Figure 1.5.1(a) Graph of p/q and its horizontal asymptote (red)
Stepwise solution
Divide p by x3, where 3 is the highest power in the denominator. Press the Enter key.
Context Panel: Expand≻Expand
Context Panel: Assign to a Name≻P
px3
4x3+5x2+6x+7x3
= expand
4+5x+6x2+7x3
→assign to a name
P
limx→∞P = 4
Divide q by x3, where 3 is the highest power in the denominator. Press the Enter key.
qx3
7x3+6x2+5x+4x3
7+6x+5x2+4x3
Q
limx→∞Q = 7≠0
Divide P by Q
PQ = 4+5x+6x2+7x37+6x+5x2+4x3
The limit of the rational function pq is the limit of PQ. Since limx→∞Q≠0, apply the Quotient rule.
limx→∞pq=limx→∞PQ=limx→∞Plimx→∞Q=47
The limit of PQ is the quotient of the limits, namely, 47. Note how the limit is the ratio of the coefficients of the leading terms in the numerator and denominator of p and q, respectively.
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