This example requires all work to be shown and justified. Thank you for any help
ID: 2874022 • Letter: T
Question
This example requires all work to be shown and justified. Thank you for any help or leads that I can be given!!
The problem:
Consider the function f(x,y) = (x^4*y)/(x^8y*x^4 + y^2)
The limit as (x, y) (0, 0) does not exist, even though the limit along any line and parabola does.
Answer the following questions.
(1) Show that the limit along any line r(t) =< at, bt > when t 0 exists and equals 0.
(2) Show that the limit along any parabola c(t) =< at, bt^2 > when t 0 exists and equals 0.
(3) Calculate f(x, y) at the points (10^1 , 10^4 ), (10^5 , 10^20), (10^20 , 10^80).
(4) Show that lim(x,y)(0,0) f(x, y) does not exist. Hint: Compute the limit along the curve y = x^4 .
Explanation / Answer
(1) Show that the limit along any line r(t) =< at, bt > when t 0 exists and equals 0.
f(x,y) = (x^4*y)/(x^8y*x^4 + y^2)
limt->0 (a4t4*b t)/(a8t8bt*a4t4 +b2t2)
limt->0 (a4bt3)/(a8t2ba4t3 +b2)
=(0)/(00 +b2)
=0
(2) Show that the limit along any parabola c(t) =< at, bt^2 > when t 0 exists and equals 0.
limt->0 (a4t4*b t2)/(a8t8bt2*a4t4 +b2t4)
limt->0 (a4bt2)/(a8t4ba4t2 +b2)
=0/(0-0+b2)
=0
(3) Calculate f(x, y) at the points (10^1 , 10^4 ), (10^5 , 10^20), (10^20 , 10^80).
f(10^1 , 10^4 ) = (10-4*10-4)/(10-8 -10-410-4+10-8)
f(10^1 , 10^4 ) = (10-8)/(0+10-8)
f(10^1 , 10^4 ) = 1
f(10^5 , 10^20 ) = (10-20*10-20)/(10-40 -10-2010-20+10-40)
f(10^5 , 10^20 ) = (10-40)/(0+10-40)
f(10^5 , 10^20) = 1
f(10^20 , 10^80) = (10-80*10-80)/(10-160 -10-8010-80+10-160)
f(10^20 , 10^80) = (10-160)/(0+10-160)
f(10^20 , 10^80) = 1
(4) Show that lim(x,y)(0,0) f(x, y) does not exist. Hint: Compute the limit along the curve y = x^4 .
y = x^4
lim(x,y)(0,0) (x4*y)/(x8y*x4 + y2)
lim(x)(0) (x4*x4 )/(x8x4*x4 + x8)
lim(x)(0) (x8)/(0 + x8)
lim(x)(0) 1
=1
along y=0
lim(x,y)(0,0) (x4*y)/(x8y*x4 + y2)
=lim(x)(0) (x4*0)/(x80*x4 + 02)
=lim(x)(0) (0)/(x8)
=lim(x)(0) 0
=0
different paths have different limit so lim(x,y)(0,0) f(x, y) does not exist
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