14. Let c E R and let ƒ : R → R be such that lim(f(x))² = L. a. Show that if L = 0, then lim f(x) = 0. T->C →C b. Show by example that if L 0, then ƒ may not have a limit at c.

Let f (x, y) = xy/(x2 + y2). Show that f (x, y) approaches zeroalong the x- and y-axes. Then prove that lim (x,y)→(0,0)f (x, y) does not exist by showing that the limit along the line y = x is nonzero.

Consider the limit limx→0 1 x 2 = ∞ and let M = 1000. Find the largest value of δ > 0 such that if x ∈ (c − δ, c) ∪ (c, c + δ), then f(x) ∈ (1000,∞). That is, now close does x have to be to 0 so that 1/x2 is in the interval (1000,∞).

Let f(x, y) = xy /(x 2 + y2). Show that f(x, y) approaches zero along the x- and y-axes. Then prove that lim f(x, y) does not exist by (x,y) (0,0)showing that the limit along the line y =xis nonzero.