Chapter 1: Limits
Section 1.5: Limits at Infinity and Infinite Limits
Graph the function fx=x3+x3−6⁢x2+12x+22 and determine all its asymptotes.
Tools≻Load Package: Student Calculus 1
Control-drag (or copy/paste) fx=…
Context Panel: Assign Function
fx=x3+x3−6⁢x2+12x+22→assign as functionf
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
Figure 1.5.9(a) Graph of fx and its asymptotes
Obtain the rules in gx
simplifyfx assuming x<0
h≔simplifyfx assuming x>0
Obtain the asymptotes
Apply the Asymptotes command from the Student Calculus1 package.
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' = 2⁢x−14
2 x−14+rx+22 = 2⁢x−14+68+48⁢xx+22= simplify 2⁢x3−3⁢x2+6x+22 = h
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