Chapter 1: Limits
Section 1.5: Limits at Infinity and Infinite Limits
Evaluate limx→∞sinxgx, where gx=x−1x2+x+1.
Figure 1.5.4(a) has a graph of sinxx−1x2+x+1 in red, and graphs of ±x−1x2+x+1 in blue and green, respectively. The blue and green curves are envelopes for the decreasing oscillations of sinx. The figure suggests that limx→∞sinxgx=0.
Unfortunately, since limx→∞sinx does not exist, the Product rule for limits cannot be applied. Indeed, since sinx≤1, Principle 1.1.1 could be invoked. Because limx→∞gx=0, it immediately follows that limx→∞sinxgx=0.
g1 := (x-1)/(x^2+x+1):
f1 := sin(x)*g1:
Figure 1.5.4(a) Graph of sinxx−1x2+x+1
Application of Maple's limit operator
Expression palette: Limit operator
A space or explicit multiplication between factors in the numerator is essential.
Context Panel: Evaluate and Display Inline
limx→∞sinx x−1x2+x+1 = 0
Application of the Squeeze theorem
With gx=x−1x2+x+1, and sinxgx≤gx because sinx≤1, it follows that
Since limx→∞gx=limx→∞−gx=0, both sides of the inequality go to zero, so the middle term must also go to zero, by the Squeeze theorem.
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