Chapter 1: Limits
Section 1.5: Limits at Infinity and Infinite Limits
Example 1.5.2
Evaluate limx→∞pxqx, where p and q are respectively, the polynomials 4 x3+5 x2+6 x+7, and 7 x4+6 x3+5 x2+4 x+3.
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 x4+6 x3+5 x2+4 x+3→assign to a nameq
Apply Maple's limit operator
Expression palette: Limit operator
Context Panel: Evaluate and Display Inline
limx→∞pq = 0
Draw a graph
Code for Figure 1.5.2(a) is hidden in the cell containing the graph.
To obtain Figure 1.5.2(a), interactively, invoke the Plot Builder on p/q. This results in the black curve shown in Figure 1.5.2(a).
The line y=0 is a horizontal asymptote
(The relevant options for the Plot Builder are: in Basic Options set the range for x, and 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^4+6*x^3+5*x^2+4*x+3; p1:=plot([0,p/q],x=0..20,y=0..1/2,color=[red,black],thickness=[3,1]): p2:=textplot([3,.03,typeset(y=0)]): p3:=display(p1,p2); print(p3); end module: end use:
Figure 1.5.2(a) Graph of p/q and its horizontal asymptote (red)
Stepwise solution
Divide p by x4, where 4 is the highest power in the denominator. Press the Enter key.
Context Panel: Expand≻Expand
Context Panel: Assign to a Name≻P
px4
4x3+5x2+6x+7x4
= expand
4x+5x2+6x3+7x4
→assign to a name
P
limx→∞P = 0
Divide q by x4, where 4 is the highest power in the denominator. Press the Enter key.
qx4
7x4+6x3+5x2+4x+3x4
7+6x+5x2+4x3+3x4
Q
limx→∞Q = 7≠0
Divide P by Q
PQ = 4x+5x2+6x3+7x47+6x+5x2+4x3+3x4
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=0
The limit of PQ is the quotient of the limits, namely, 0.
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