#1) f(x)=(x^2+x-12)/(x^2-x-6)
#1) prove that the horizontal asymptote is y=1 by long division.
f(x)=(x^2+x-12)/(x^2-x-6)
show(f(x))
show(f(x).partial_fraction())
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#2) f(x)=(x^2+x-12)/(x^2-x-6)
#2) why does f(x) only have 1 vertical asymptote?
show(factor(x^2+x-12))
show(factor(x^2-x-6))
show(factor((x^2+x-12)/(x^2-x-6)))
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#3) f(x)=(x^2+x-12)/(x^2-x-6)
#3) show that the intercepts of f(x) are (-4,0) and (0,2).
show(solve(f(x)==0,x))
show(f(0))
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#4) f(x)=(x^2+x-12)/(x^2-x-6)
#4) make a complete sketch of f(x) labeling all intercepts and asymptotes.
p1=plot(f(x),-10,-2,color='red')
p2=plot(f(x),-2,3,color='green')
p3=plot(f(x),3,10,color='purple')
p4=plot(1,-10,10)
(p1+p2+p3+p4).show(ymin=-5,ymax=5)
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#5) f(x)=(x^2+x-12)/(x^2-x-6)
#5) state the domain and range of f(x).
#5) domain: (-inf,-2)U(-2,3)U(3,+inf)
#5) range: (-inf,1)U(1,+inf)
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#6) f(x)=(x^2+x-12)/(x^2-x-6)
#6) discuss the important limits
show(limit(f(x),x=-infinity))
show(limit(f(x),x=-2,dir='left'))
show(limit(f(x),x=-2,dir='right'))
show(limit(f(x),x=3,dir='left'))
show(limit(f(x),x=3,dir='right'))
show(limit(f(x),x=infinity))
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