Approaching the Limit: Solving a Tricky Exam Problem

[thepatient]

limit problem from past exam. :(

Homework Statement

Problem from an exam that I just did is really bugging me.

lim(x,y)->(0,0) (x*y²)/(x²+y⁴)
The attempt at a solution[/b]
At first I thought the limit didn't exist. I tried using paths, x=y^1/2, but then I couldn't find a counter example for which a path approached at a different point. So then I tried using the squeeze theorem, and got that the limit is 0. :\

Was my approach to the problem correct? is the limit really 0? [

 

[Dick]

You didn't say how you proved it. But yes, the limit is zero.

 

[thepatient]

You didn't say how you proved it. But yes, the limit is zero.

Oh really? XD Well I assumed:

0<y²<x²+y⁴ (since x^2 and y^4 are always positive, and y²<y⁴

0<y²/x²+y⁴<1
Then multiplying x into the inequality:

0<x*y²/x²+y⁴<x

Then taking the limit:
lim(x,y)->(0,0) 0<lim(x,y)->(0,0) x*y²/x²+y⁴<lim(x,y)->(0,0) x

Which gave me that:
0<lim(x,y)->(0,0) x*y²/x²+y⁴<0

which by the squeeze theorem, limit is zero. Was that correct...?

 

[Dick]

Oh really? XD Well I assumed:

0<y²<x²+y⁴

0<y²/x²+y⁴<1
Then multiplying x into the inequality:

0<x*y²/x²+y⁴<x

Then taking the limit:
lim(x,y)->(0,0) 0<lim(x,y)->(0,0) x*y²/x²+y⁴<lim(x,y)->(0,0) x

Which gave me that:
0<lim(x,y)->(0,0) x*y²/x²+y⁴<0

which by the squeeze theorem, limit is zero. Was that correct...?

Well the starting point 0<y^2<x^2+y^4 is wrong. Suppose x=0 and y=(1/2). I would use polar coordinates.

 

[tt2348]

Limit doesn't exist , try y=x^(1/2)

 

[thepatient]

Aaah... I was starting it with polar coordinates at first and it seemed like the best way to go, but then it seemed to messy so I thought I was doing it wrong. I was running out of time and just left it like that. XD Thanks lots, at least maybe I have partial credit haha..

 

[tt2348]

lim x->0 f(x,x^(1/2))=1/2
and lim x->0 f(x,0)=0

 

[Dick]

Limit doesn't exist , try y=x^(1/2)

Ooops. You are so right. Sorry.

 

[thepatient]

Aww darn... it was a 3 part question worth 5 points total on that part of the exam, so I should have 10/3 credit hehe... The rest of the test was pretty good.