Limits — Question 6

Question 6 – Solution

Strategy. A relative limit uses only paths in the domain, but all such paths must still agree.

Step 1: Directions The wedge is $0\le\theta\le\pi/3$. In polar form, $$f=\frac{r^2\cos^2\theta-r^2\sin^2\theta}{r^2}=\cos 2\theta.$$

Step 2: Boundary paths Along $y=0$ ($\theta=0$), $f=1$. Along $y=\sqrt 3x$ ($\theta=\pi/3$), $$f=\cos(2\pi/3)=-\frac 12.$$ Both paths stay in $D$ and approach the origin.

Conclusion. Because the two permitted approaches disagree, $$\begin{gathered}\boxed{\displaystyle \lim_{\substack{(x,y)\to(0,0)\\(x,y)\in D}}f(x,y)\text{ does not exist}.}\end{gathered}$$