Answer to Question #241843 in Calculus for Sakshi

Question #241843

Let D = {(x, y) ∈ R2 : x > 0, 0 < y < x3}. Define

f(x, y) = (0, (x, y)not belong to D

1, (x, y) ∈ D.


a) Approaching (0, 0) along the line y = mx for each real number m and the y-axis, prove that

lim(x,y)→(0,0) f(x, y) exists and compute the limit.

b) Argue whether f is continuous at (0, 0).


1
Expert's answer
2021-09-28T11:56:09-0400


(a)

"\\lim\\limits_{(x,y)\\rightarrow (0,0)} f(x,y)\\\\\n=\\lim\\limits_{(x,y)\\rightarrow (0,0)} f(x,mx)\\\\\nSo, \\lim\\limits_{(x,y)\\rightarrow (0,0)} f(x,0)\\leq\\lim\\limits_{(x,y)\\rightarrow (0,0)} f(x,y)\\leq\\lim\\limits_{(x,y)\\rightarrow (0,0)} (mx)^3\\\\\n\\Rightarrow 0\\leq\\lim\\limits_{(x,y)\\rightarrow (0,0)} f(x,y)\\leq0\\\\\nHence, \\lim\\limits_{(x,y)\\rightarrow (0,0)} f(x,y)=0"

(b) Yes, the function is continuous at "x=0."

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