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Multivariable Calculus - a question of limits

  • #1

Homework Statement


Prove the following, using the meaning of a limit:
2nsxd29.png


Homework Equations


epsilon > 0, delta > 0

0 < sqrt(x^2 + y^2) < delta
| f(x) - 0 | < epsilon (1)

The Attempt at a Solution



So, I know that I have to elaborate on the inequality in (1), further. However, I'm not sure of how to go about this. If I start transposing it seems to get much too messy and I end up with a proposed delta which seems far too complex. Is there some simplification that I'm overlooking here?

Any help is really appreciated, ta.
 

Answers and Replies

  • #2
12
0
Could you not use polar coordinates instead?
Then the equation would be much simpler and the limit is just as r goes to 0
 
  • #3
I did consider that but I'm pretty sure the limit definition must be used here (as a requirement).

Does anyone have any ideas of how to simplify this beast?

EDIT: Really need this one clarified guys. Does anyone have any hints at all? Even a vague idea.
 
  • #4
vela
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Hint: use the fact that |x|<r and |y|<r, where r=sqrt(x^2+y^2).
 
  • #5
Hint: use the fact that |x|<r and |y|<r, where r=sqrt(x^2+y^2).
Bingo, great hint, thank you! Subtle yet it serves me well :D♦
 
Last edited:

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