Calculus UNIT201 2-sided limits

2827 days ago by MATH4R2013

#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) 
       








#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) 
       














#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))