Chapter 7. Asymptotes of Curves

Rovenski Vladimir, Haifa

7.1 Asymptotes of Curves

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Example 1 . Derive and plot asymptotes of the graph

a) y= , b) y= , c) y= .

Answer : a) two-sided asymptote y= x.

> f:=surd(x^3+2, 3): x1:=2:

> k:=limit(f/x, x=infinity); b:=limit(f-k*x, x=infinity);

> plot([f, k*x+b], x=-x1..x1, thickness=[2,1], linestyle=[1,2]);

b) Answer : the two-sided asymptote .

> f:=surd(6*x^2-x^3, 3):

> k:=limit(f/x, x=-infinity); b:=limit(f-k*x, x=-infinity);

> plot([f, k*x+b], x=-6..10, thickness=[2,1], linestyle=[1,2]);

c) Answer : vertical and oblique asymptotes are the following:

> f:=x^3/(2*(x+1)^2): k:=limit(f/x, x=infinity); b:=limit(f-k*x, x=infinity);

> p||1:=plot(f, x=-0.84..5, thickness=2): p||2:=plot(f, x=-6..-1.3, thickness=2): p||3:=plot(k*x+b, x=-6..5, linestyle=2): p||4:=plot([-1, x, x=-12..1.8], linestyle=2):

> plots[display]([seq(p||i, i=1..4)]);

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Example 2 . Derive and plot asymptotes of the parameterized curve .

Answer : The curve has four points running to infinity. Its three asymptotes are .

> X:=t^2/(1-t): Y:=t^3/(1-t^2); N:=`union`({-6..-1.15}, {-0.9..0.9}, {1.15..6});:

> p1:=plot([seq([X,Y, t=N[i]], i=1..3)], thickness=2): %;

> A:=[-infinity, -1, 1, infinity]: for i from 1 to nops(A) do k||i:=limit(Y/X, t=A[i]): b||i:=limit(Y-k||i*X, t=A[i]) od;

> p2:=plot([seq(k||i*x+b||i, i=[1,3,4]), [subs(t=A[2],X), x, x=-6..8]], x=-7..7, linestyle=2):%;

> plots[display]([p1, p2], scaling=constrained);

Example 3 . Plot the space curve and find its three asymptotes (coordinate axes):

> f:=exp(-t^2): r:=[f*(t-2)*(t-1)/t, f*(t-2)*t/(t-1), f*t*(t-1)/(t-2)];

> r := [exp(-t^2)*(t-2)*(t-1)/t, exp(-t^2)*(t-2)*t/(t-1), exp(-t^2)*t*(t-1)/(t-2)];

> plots[spacecurve](r, t=-1...1.9, thickness=3,axes=normal);

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