(a) In polar coordinates
x 3 − y 3 x 2 + y 2 = r 3 cos 3 θ − r 3 sin 3 θ r 2 = r cos 3 θ − r sin 3 θ \frac{x^3-y^3}{x^2+y^2}=\frac{r^3\cos^3\theta-r^3\sin^3\theta}{r^2}
=r\cos^3\theta-r\sin^3\theta x 2 + y 2 x 3 − y 3 = r 2 r 3 cos 3 θ − r 3 sin 3 θ = r cos 3 θ − r sin 3 θ
Since
∣ r cos 3 θ − r sin 3 θ ∣ ≤ 2 r \big|r\cos^3\theta-r\sin^3\theta\big|\le 2r r cos 3 θ − r sin 3 θ ≤ 2 r
and 2 r → 0 2r\rightarrow 0 2 r → 0 r → 0 r\rightarrow 0 r → 0 0 0 0
(b)
The limit as we approach ( 0 , 0 ) (0,0) ( 0 , 0 ) x x x
lim t → 0 x 2 − y 4 x 2 + y 4 ∣ ( x , y ) = ( t , 0 ) = lim t → 0 t 2 − 0 4 t 2 + 0 4 = 1 \begin{aligned}
\lim_{t\rightarrow 0}\frac{x^2-y^4}{x^2+y^4}\bigg|_{(x,y)=(t,0)}
=\lim_{t\rightarrow 0}\frac{t^2-0^4}{t^2+0^4}
=1
\end{aligned} t → 0 lim x 2 + y 4 x 2 − y 4 ( x , y ) = ( t , 0 ) = t → 0 lim t 2 + 0 4 t 2 − 0 4 = 1
On the other hand the limit as we approach ( 0 , 0 ) (0,0) ( 0 , 0 ) y y y
lim t → 0 x 2 − y 4 x 2 + y 4 ∣ ( x , y ) = ( 0 , t ) = lim t → 0 0 2 − t 4 0 2 + t 4 = − 1 \begin{aligned}
\lim_{t\rightarrow 0}\frac{x^2-y^4}{x^2+y^4}\bigg|_{(x,y)=(0,t)}
=\lim_{t\rightarrow 0}\frac{0^2-t^4}{0^2+t^4}
=-1
\end{aligned} t → 0 lim x 2 + y 4 x 2 − y 4 ( x , y ) = ( 0 , t ) = t → 0 lim 0 2 + t 4 0 2 − t 4 = − 1
These are different, so the limit as ( x , y ) → 0 (x,y)\rightarrow 0 ( x , y ) → 0
We can gain a more detailed understanding of the behaviour of
x 2 − y 4 x 2 + y 4 \frac{x^2-y^4}{x^2+y^4} x 2 + y 4 x 2 − y 4
x 2 − y 4 x 2 + y 4 = r 2 cos 2 θ − r 4 sin 4 θ r 2 cos 2 θ + r 4 sin 4 θ = cos 2 θ − r 2 sin 4 θ cos 2 θ + r 2 sin 4 θ \frac{x^2-y^4}{x^2+y^4}
=\frac{r^2\cos^2\theta-r^4\sin^4\theta}{r^2\cos^2\theta+r^4\sin^4\theta}
=\frac{\cos^2\theta-r^2\sin^4\theta}{\cos^2\theta+r^2\sin^4\theta} x 2 + y 4 x 2 − y 4 = r 2 cos 2 θ + r 4 sin 4 θ r 2 cos 2 θ − r 4 sin 4 θ = cos 2 θ + r 2 sin 4 θ cos 2 θ − r 2 sin 4 θ
Now fix any θ \theta θ r → 0 r\rightarrow 0 r → 0 θ \theta θ x x x cos θ ≠ 0 \cos\theta\ne 0 cos θ = 0 y y y
lim r → 0 cos 2 θ − r 2 sin 4 θ cos 2 θ + r 2 sin 4 θ = cos 2 θ cos 2 θ = 1 \begin{aligned}
\lim_{r\rightarrow 0}
\frac{\cos^2\theta-r^2\sin^4\theta}{\cos^2\theta+r^2\sin^4\theta}
=\frac{\cos^2\theta}{\cos^2\theta}
=1
\end{aligned} r → 0 lim cos 2 θ + r 2 sin 4 θ cos 2 θ − r 2 sin 4 θ = cos 2 θ cos 2 θ = 1
But if cos θ = 0 \cos\theta= 0 cos θ = 0 y y y
lim r → 0 cos 2 θ − r 2 sin 4 θ cos 2 θ + r 2 sin 4 θ = lim r → 0 − r 2 sin 4 θ r 2 sin 4 θ = − sin 4 θ sin 4 θ = − 1 \begin{aligned}
\lim_{r\rightarrow 0}
\frac{\cos^2\theta-r^2\sin^4\theta}{\cos^2\theta+r^2\sin^4\theta}
=\lim_{r\rightarrow 0}
\frac{-r^2\sin^4\theta}{r^2\sin^4\theta}
=\frac{-\sin^4\theta}{\sin^4\theta}
=-1
\end{aligned} r → 0 lim cos 2 θ + r 2 sin 4 θ cos 2 θ − r 2 sin 4 θ = r → 0 lim r 2 sin 4 θ − r 2 sin 4 θ = sin 4 θ − sin 4 θ = − 1