Question

PLEASE HELP ME :( prove limit of (x^3*y)/(sqrt(x^8+y^8) does not exist

Answer 1

limit as x,y approach ?

you essentially have to approach it from all directions and paths

Answer 2

yes, I have tried from x=0, y=0, y=mx, etc. I cannot find an equation where it will not approach 0 as the limit

Answer 3

it is the limit as (x,y) approach (0,0)

Answer 4

\[\large\frac{x^3y}{\sqrt{x^8+y^8~}}\]
hmm, can you do LHop on it?

Answer 5

I'm not sure how that would work in this type of problem

Answer 6

me either ... something to do with Fx Fy Fxy Fyx perhaps

Answer 7

try a quadratic approach? y = m(x-a)^2+b

Answer 8

of course the +b is prolly 0 for that ...

Answer 9

the limit of y=x is 1/sqrt(2)
the limit of y=2x is 2/sqrt(257)

the limit is different depending on the path taken ...

Answer 10

different limits means no limit can be defined

Answer 11

let y = mx

\[\large\frac{mx^4}{x^4\sqrt{1+m^8}}\]
\[\lim_{x\to 0} \large\frac{m}{\sqrt{1+m^8}}\]is not the same for all m

Answer 12

when I did the limit as x approaches 0 with y=mx I got 0, was that wrong?

Answer 13

im going to go out on a limb and say .... yes, you were wrong :)

Answer 14

the xs cancel out, leaving us a limit that differs from m to m

Answer 15

how did you get the limit of y=x is 1/sqrt(2)
the limit of y=2x is 2/sqrt(257), I am getting the limit with y=x as x approaches 0 to equal 1/sqrt(x^4) which, when x = 0 wouldn't give you that?

Answer 16

for y = mx

\[\frac{m}{\sqrt{1+m^8}}\]
\[\frac{1}{\sqrt{1+1}}\]
\[\frac{2}{\sqrt{1+2^8}}\]
\[\frac{3}{\sqrt{1+3^8}}\]
for any given direction of m, we get a different limit value

Answer 17

how are you getting from mx^4/sqrt(x^8+mx^8) to that original part?

Answer 18

mx^4/sqrt(x^8(1+m^8)) then what, you can factor out x^4??

Answer 19

the most basic definition of a limit is that it must approach the same value regardless of direction ... otherwise it does not exist or cannot be calculated
\[\large\frac{x^3y}{\sqrt{x^8+y^8~}}\]
\[y=mx\]
\[\large\frac{x^3(mx)}{\sqrt{x^8+(mx)^8~}}\]
\[\large\frac{mx^4}{\sqrt{x^8+m^8x^8~}}\]
\[\large\frac{mx^4}{\sqrt{x^8(1+m^8)~}}\]
\[\large\frac{mx^4}{\sqrt{x^8}\sqrt{1+m^8~}}\]
\[\large\frac{mx^4}{{x^{8/2}}\sqrt{1+m^8~}}\]
\[\large\frac{m\cancel{x^4}}{\cancel{x^{4}}\sqrt{1+m^8~}}\]

Answer 20

the limit is not the same for all values of m, therefore it DNE or cannot be determined

Answer 21

THANK YOU so much!! I have been having trouble with this problem for the past week and it's due with a problem set tomorrow... I really appreciate your help!!