Central axis of the hyperboloid

Question

Question 1: Give a complete description (in words) of the surface (x - 1)² + y² + 4(z- 6)²= 16.

Question 2:

(a) Build an equation for a hyperboloid of two sheets with the following properties:

-The central axis of the hyperboloid is the x-axis.

-The two sheets are 12 units apart (at closest), and are mirror images of each other across the yz-plane.

-The hyperboloid intersects the plane x = 10 in a circle of radius 4.

(b) Find a function y = f(x) whose graph in the xy-plane, when rotated around the x-axis, would produce the hyperboloid built in (a).

Question 3: Define f(x, y) = (x³y)/(x⁶+ y²)
                                       
Investigate limf(x, y).
                (x,y) → (0,0)

(a) Show that the limit is zero when (0,0) is approached along any line. (Consider x = 0 separately from y = mx.)

(b) Show that the limit is zero when (0,0) is approached along any parabola of the form y = kx².

(c) Find an approach to (0,0) where the limit is non-zero, thereby demonstrating that the overall limit doesn’t exist.