> restart;# Don't do restart on web maple, it always restarts (screws up plots)
> MyName:="The good doctor";Date:="Sep 11, 2001";
> #limits, infinity, and turtles
> f:=sin(x)/x;
> eqn:=Limit(f,x=0)=limit(f,x=0);
> f:=x^2*exp(-x);
> expression:=Limit(f,x=infinity)=limit(f,x=infinity);
> f:=sqrt(x+1)-sqrt(x);
> Limit(f,x=infinity)=limit(f,x=infinity);
> infinity-infinity;
> 0/0;
> infinity/infinity;
> 0*infinity;
> #L'H
> f:='f'; Limit(f,x=a)=0, Limit(g,x=a)= 0; Limit(f/g,x=a)=Limit(Diff(f,x)/Diff(g,x),x=a);
> Limit(sin(2*x)/tan(3*x),x=0);
> Limit(sin(2*x),x=0)=limit(sin(2*x),x=0);val:=subs({x=0},sin(2*x));evalf(val);
> Limit(tan(3*x),x=0)=limit(tan(3*x),x=0);val:=subs({x=0},tan(3*x));evalf(val);
> val:=subs({x=0},sin(2*x)/tan(3*x));evalf(val);
> val:=subs({x=0},diff(sin(2*x),x)/diff(tan(3*x),x));eval(val);
> diff(sin(2*x),x);diff(tan(3*x),x);Diff(sin(2*x),x)=diff(sin(2*x),x);
> 
> limit(diff(sin(2*x),x),x=0);limit(diff(tan(3*x),x),x=0);
> #Why this works -- make L `small'
> L:=2*Pi;plot(sin(2*x),x=-L..L,title="sin(2x)");
> L:=Pi/8;plot(tan(3*x),x=-L..L,title="tan(3x)");
> L:=0.1;plot([sin(2*x),tan(3*x)],x=-L..L,color=[red,blue],title="both",legend=["sin 2x", "tan 3x"]);
> #improper 
> int(1/x^2,x=1..infinity);Int(1/x^2,x=1..infinity)=int(1/x^2,x=1..infinity);
> int(1/x^2,x);
> int(1/x^2,x=1..M);
> int(1/x,x=1..M);
> int(1/x^1.001,x=1..M);
> int(1/x^0.999,x=1..M);
> # p-series
> with(student);
> leftsum(2^(-x),x=0..n,n);
> leftbox(2^(-x),x=0..10,10);
> rightsum(2^(-x),x=0..n,n);
> rightbox(2^(-x),x=0..10,10);
> #repeat for f:=1/x
> #topologists sin curve and numpoints