Chapter 1: Limits
Section 1.5: Limits at Infinity and Infinite Limits
Example 1.5.9
Graph the function fx=x3+x3−6x2+12x+22 and determine all its asymptotes.
Solution
Initialize
Tools≻Load Package: Student Calculus 1
Loading Student:-Calculus1
Control-drag (or copy/paste) fx=…
Context Panel: Assign Function
fx=x3+x3−6x2+12x+22→assign as functionf
Graph fx
Because of the absolute value in its numerator, fx is not a rational function. That's why, in Figure 1.5.9(a), its graph shows, in addition to a vertical asymptote, an oblique asymptote only on the right, and a horizontal asymptote only on the left.
The function can be represented as the piecewise rational
gx={−6x2−2x+22x<02x3−3x2+6x+22x>0
ff:=(x^3+abs(x)^3-6*x^2+12)/ (x+2)^2: p1:=plot(ff,x=-20..20,y=-50..30, discont=true,color=black): p2:=plot([[x,2*x-14,x=5..20],[x,-6,x=-20..-4]],color=red,linestyle=dash): p3:=plots:-implicitplot(x=-2, x=-20..20,y=-50..30,color=red,linestyle=dash): plots:-display([p1,p2,p3]);
Figure 1.5.9(a) Graph of fx and its asymptotes
Obtain the rules in gx
simplifyfx assuming x<0
−6x2−2x+22
h≔simplifyfx assuming x>0
2x3−3x2+6x+22
Obtain the asymptotes
Apply the Asymptotes command from the Student Calculus1 package.
Asymptotesfx,x
y=2x−14,y=−6,x=−2
That x=−2 is the only zero of the denominator of fx is obvious by inspection. The numerator reduces to −6x2−2 when x<0, and evaluates to −12≠0 at x=−2. Hence, x=−2 is the equation of the only vertical asymptote.
Determine the asymptotes from first principles:
Obtain the horizontal asymptote y=−6
Expression palette: Limit operator
limx→−∞fx = −6
Obtain the vertical asymptote x=−2
limx→−2fx = −∞
Obtain the oblique asymptote y=2 x−14
Apply long division to h via the quo command. The remainder is assigned to r.
quonumerh,denomh,x,'r' = 2x−14
2 x−14+rx+22 = 2x−14+68+48xx+22= simplify 2x3−3x2+6x+22 = h
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