Chapter 1: Limits
Section 1.5: Limits at Infinity and Infinite Limits
Find the horizontal asymptote(s) for fx=3⁢x5+1x5+5⁢x2+1+10⁢sin⁡xx2+1.
Define the function f
Control-drag (or type) fx=…
Context Panel: Assign Function
fx=3⁢x5+1x5+5⁢x2+1+10⁢sin⁡xx2+1→assign as function
Figure 1.5.5(a), a graph of fx, suggests that the lines y=±3 are horizontal asymptotes.
Notice that the graph of fx approaches the line y=3 only for large positive values of x, and that the graph of fx approaches the line y=−3 for only for x→−∞.
plot([(3*x^5+1)/(abs(x)^5+5*x^2+1)+10*sin(x)/(x^2+1),3,-3],x=-20..20,y=-5..5, color=[red,blue,green],thickness=3,legend =[typeset('f'(x)),typeset(y=3),typeset(y = -3)]);
Figure 1.5.5(a) Graph of fx in red, and asymptotes y=±3 in blue and green, respectively
Application of Maple's limit operator
Expression palette: Limit operator
Context Panel: Evaluate and Display Inline
limx→∞fx = limx→∞f⁡x
limx→−∞fx = limx→−∞f⁡x
Solution from first principles
Application of the Sum rule is valid if the limit of each summand exists. There are two summands:
gx=3⁢x5+1x5+5⁢x2+1 and hx=10⁢sin⁡xx2+1
The limits at ±∞ for hx are handled by Principle 1.1.1 or the Squeeze theorem, as per Example 1.5.4. These limits will certainly be zero.
In evaluating the limit at +∞ for gx, the absolute value in the denominator can be ignored because x>0 for large x. Hence, the limit is that of a rational function in which both numerator and denominator have the same degree. The limit will be the ratio of the leading coefficients of the numerator and denominator, that is, 3.
In evaluating the limit at −∞ for gx, the absolute value in the denominator is replaced by −x, so that again, the limit is taken of a rational function, namely, the rational function
Dividing the numerator and denominator of this fraction by x5, the degree of the denominator, leads to
from which the limit −3 is readily obtained.
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