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azure kitsune
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Homework Statement
Let
[tex]f(x,y) = \dfrac{x^2+2xy^2+y^2}{x^2+y^2}[/tex]
Prove that
[tex]\lim_{(x,y) \to (0,0)} f(x,y) = 1[/tex]
Homework Equations
Definition of the limit of a function of multiple variables:
It suffices to show that for all [tex]\epsilon > 0[/tex], there exists a [tex]\delta > 0[/tex] such that for all [tex](x,y)[/tex] such that [tex]0 < x^2 + y^2 < \delta ^2[/tex], we have [tex]|f(x,y) - 1| < \epsilon[/tex]
The Attempt at a Solution
[tex]|f(x,y) - 1| = \left| \dfrac{2xy^2}{x^2+y^2} \right| = \dfrac{2|x|y^2}{x^2+y^2} [/tex]
I need to bound this with an expression in terms of [tex]\delta[/tex], but I can't think of any way to do so. I noticed that the denominator is less than [tex]\delta ^2[/tex] but I can't get anywhere with that. (I end up bounding it in the wrong direction! )
Can anyone point me in the right direction? Thanks.
[Edit: Good catch Mark44!]
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