Suppose f : R+ R is continuous and f(0) = 0. Consider the limit yf(x) lim (x,y) (0,0) y² + f (x)2* Does this exist for all such f, not exist for any such f, or exist for some such f? Explain.
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- 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.Let f : [1, ∞) → R be a continuous function such that lim|f(x)| = 0. For n ≥ 1, let gn : [1, ∞) → R be given by gn(x) = f(nx). Show that {gn}n≥1 is equicontinuous on [1, ∞).Let ƒ: S→R be an uniformly continuous function. Finish the proof by showing that the limit lim_(x →b)ƒ(x) exists.
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