University Calculus: Early Transcendentals (3rd Edition)

$1$
Use polar-coordinates: $x= r \cos \theta , y = r \sin \theta \\r^2=x^2+y^2$ Now, $\lim\limits_{(x,y) \to (0,0) } f(x,y)=\lim\limits_{(x,y) \to (0,0) } \cos \dfrac{x^3-xy^2}{x^2+y^2} \\=\lim\limits_{r \to 0} \cos (\dfrac{ r^3 \cos^3 \theta- r^3 \sin^3 \theta}{r^2}) \\\lim\limits_{r \to 0} \cos [r^3 (\dfrac{ \cos^3 \theta- \sin^3 \theta}{r^2}] \\=\cos 0 \\=1$