Chapter 1: Limits
Section 1.5: Limits at Infinity and Infinite Limits
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Example 1.5.2
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Evaluate , where and are respectively, the polynomials , and .
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Solution
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Enter the data
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Control-drag (or type) .
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Context Panel: Assign to a Name≻
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Control-drag (or type) .
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Context Panel: Assign to a Name≻
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Apply Maple's limit operator
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Expression palette: Limit operator
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Context Panel: Evaluate and Display Inline
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Draw a graph
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Code for Figure 1.5.2(a) is hidden in the cell containing the graph.
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To obtain Figure 1.5.2(a), interactively, invoke the Plot Builder on . This results in the black curve shown in Figure 1.5.2(a).
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The line is a horizontal asymptote
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(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].)
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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:
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Figure 1.5.2(a) Graph of and its horizontal asymptote (red)
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Stepwise solution
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Divide by , where 4 is the highest power in the denominator.
Press the Enter key.
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Context Panel: Expand≻Expand
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Context Panel: Assign to a Name≻
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Expression palette: Limit operator
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Context Panel: Evaluate and Display Inline
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Divide by , where 4 is the highest power in the denominator.
Press the Enter key.
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Context Panel: Expand≻Expand
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Context Panel: Assign to a Name≻
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Expression palette: Limit operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Evaluate and Display Inline
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The limit of the rational function is the limit of . Since , apply the Quotient rule.
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The limit of is the quotient of the limits, namely, .
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