Can you please help me doing these questions. PLEASE SHOW EACH AND EVERY STEP WH
ID: 2874130 • Letter: C
Question
Can you please help me doing these questions. PLEASE SHOW EACH AND EVERY STEP WHILE SOLVING THESE PROBLEMS.
The problem: Consider the function F(x,y) = x^4y / X^8 - yx^4 + y^2 This is an interesting example where the limit as (x, y)rightarrow (0,0) does not exist, even though the limit along any line and parabola does. Answer the following questions. Show that the limit along any line r(t) = when t rightarrow 0 exists and equals 0. Show that the limit along any parabola c(t) = when t rightarrow 0 exists and equals 0. Calculate f(x, y) at the points (10-1,10-4), (10-5,10-20), (10-20,10-80). Do not use a calculator. Show that lim(x,y)rightarrow (0, 0) f(x,y) does not exist.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