#1) Piecewise Defined Functions!
f1(x)=x+3
f2(x)=-2*x-2
f3(x)=x^2-1
f4(x)=sqrt(1-(x-2)^2)
f=piecewise([[(-4,-2),f1],[(-2,-1),f2],[(-1,1),f3],[(1,3),f4]])
show(lim(f1(x),x=-2,dir='left'))
show(lim(f2(x),x=-2,dir='right'))
show(lim(f2(x),x=-1,dir='left'))
show(lim(f3(x),x=-1,dir='right'))
show(lim(f3(x),x=1,dir='left'))
show(lim(f4(x),x=1,dir='right'))
show(lim(f1(x),x=-4,dir='right'))
show(lim(f4(x),x=3,dir='left'))
plot(f,aspect_ratio=1)
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#2) Rational Functions
f(x)=(x^2+2*x+4)/(x+2)
show(f(x).partial_fraction())
show(lim(f(x),x=-2))
p1=plot(f(x),-5,-2)
p2=plot(f(x),-2,5)
show(p1+p2)
f(x)=(x^2+4*x+4)/(x+2)
show(f(x).partial_fraction())
show(lim(f(x),x=-2))
p1=plot(f(x),-5,-2)
p2=plot(f(x),-2,5)
show(p1+p2)
f(x)=(x^3+2*x+4)/(x+2)
show(f(x).partial_fraction())
show(lim(f(x),x=-2))
p1=plot(f(x),-25,-2)
p2=plot(f(x),-2,25)
show(p1+p2)
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#3) Limits at Infinity!
f(x)=(2*x+1)/(3*x^3-2)
show(f(x))
show(lim(f(x),x=infinity))
show(plot(f(x),1,10))
f(x)=(2*x^2+1)/(3*x-2)
show(f(x).partial_fraction())
show(lim(f(x),x=infinity))
show(plot(f(x),1,10))
f(x)=(2*x+1)/(3*x-2)
show(f(x).partial_fraction())
show(lim(f(x),x=infinity))
show(plot(f(x),1,100))
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