Limits — Question 10

Question 10 – Solution

Strategy. Recognize the cubic numerator’s polar form and isolate one factor of radial distance.

Step 1: Convert With $x=r\cos\theta$ and $y=r\sin\theta$, $$x^3-3xy^2=r^3(\cos^3\theta-3\cos\theta\sin^2\theta)=r^3\cos 3\theta.$$ Since $x^2+y^2=r^2$, for $r>0$, $$F(x,y)=r\cos 3\theta.$$

Step 2: Uniform estimate For every angle, $$|F(x,y)|=r|\cos 3\theta|\le r=\sqrt{x^2+y^2}\longrightarrow 0.$$ Hence $$\boxed{\displaystyle\lim_{(x,y)\to(0,0)}F(x,y)=0}.$$

Step 3: Continuity value Continuity requires the assigned value to equal the limit, so the unique choice is $$\boxed{c=0}.$$ The bound is independent of $\theta$, verifying that no directional or curved approach can change the result.