In this problem we show that the function lim (x,2x)→(0,0) does not have a limit as (x,y) → (0,0). (a) Suppose that we consider (x,y) → (0,0) along the curve y = 2x-y x+y f(x, y) = lim (x,3x)→(0,0) 2x - y x + y (b) Now consider (x, y) → (0,0) along the curve y = 3x. Find the limit in this case: 2x-y x+y -1 2x. Find the limit in this case: lim (x,mx)→(0,0) (Be sure that (c) Note that the results from (a) and (b) indicate that f has no limit as (x,y) → (0,0) (be sure you can explain why!). To show this more generally, consider (x,y) → (0, 0) along the curve y = mx, for arbitrary m. Find the limit in this case: 2x-y x+y you can explain how this result also indicates that f has no limit as (x, y) → (0,0).
In this problem we show that the function lim (x,2x)→(0,0) does not have a limit as (x,y) → (0,0). (a) Suppose that we consider (x,y) → (0,0) along the curve y = 2x-y x+y f(x, y) = lim (x,3x)→(0,0) 2x - y x + y (b) Now consider (x, y) → (0,0) along the curve y = 3x. Find the limit in this case: 2x-y x+y -1 2x. Find the limit in this case: lim (x,mx)→(0,0) (Be sure that (c) Note that the results from (a) and (b) indicate that f has no limit as (x,y) → (0,0) (be sure you can explain why!). To show this more generally, consider (x,y) → (0, 0) along the curve y = mx, for arbitrary m. Find the limit in this case: 2x-y x+y you can explain how this result also indicates that f has no limit as (x, y) → (0,0).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:In this problem we show that the function
lim
(x,2x)→(0,0)
does not have a limit as (x,y) → (0,0).
(a) Suppose that we consider (x, y) (0,0) along the curve y =
→
2x-y
x+y
f(x, y) =
lim
(x,3x)→(0,0)
2x - y
x + y
(b) Now consider (x, y) → (0,0) along the curve y = 3x. Find the limit in this case:
2x-y
x+y
-1
2x. Find the limit in this case:
lim
(x,mx) (0,0)
(Be sure that
(c) Note that the results from (a) and (b) indicate that f has no limit as (x, y) → (0,0) (be sure you can explain
why!).
To show this more generally, consider (x,y) → (0, 0) along the curve y = mx, for arbitrary m. Find the limit
in this case:
2x-y
x+y
you can explain how this result also indicates that f has no limit as (x, y) → (0,0).
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