Solution Found!
Explain why or why not Determine whether the
Chapter 9, Problem 47E(choose chapter or problem)
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) If the limits \(\lim _{(x, 0) \rightarrow(0,0)} f(x, 0)\) and \(\lim _{(0, y) \rightarrow(0,0)} f(0, y)\) exist and equal L, then \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)=L\).
(b) If \(\lim _{(x, y) \rightarrow(a, b)} f(x, y)=L\), then f is continuous at (a, b).
(c) If f is continuous at (a, b), then \(\lim _{(x, y) \rightarrow(a, b)} f(x, y)\) exists.
(d) If P is a boundary point of the domain of f, then P is in the domain of f.
Questions & Answers
QUESTION:
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) If the limits \(\lim _{(x, 0) \rightarrow(0,0)} f(x, 0)\) and \(\lim _{(0, y) \rightarrow(0,0)} f(0, y)\) exist and equal L, then \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)=L\).
(b) If \(\lim _{(x, y) \rightarrow(a, b)} f(x, y)=L\), then f is continuous at (a, b).
(c) If f is continuous at (a, b), then \(\lim _{(x, y) \rightarrow(a, b)} f(x, y)\) exists.
(d) If P is a boundary point of the domain of f, then P is in the domain of f.
ANSWER:Solution 47E Step 1 :1. If the limits exist and equal L .Then This statement is true .