solving with delta -epsilon

[unkempt pine]

Hello guys, i need to solve this exercise with delta epsilon for an assignment for uni. can someone pls try to solve it this way for me, because i dont understand

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[runic halo]

Fix eps >0. Can you find delta > 0 such that |f(x,y)|< eps for all (x,y) satisfying sqrt(x^2+y^2)< delta

@unkempt pine this would be by definition

[unkempt pine]

This was another example

But i can’t figure it out could you please provide the answer?

[runic halo]

there is no "the answer"

$$ \frac{|x(2x^2+xy)|}{x^2+y^2} < \varepsilon $$

[lethal fossil]

aL

[runic halo]

can you fiddle around with this expression?

for instance
$$ \frac{|x(2x^2+xy)|}{x^2+y^2} = \frac{x^2}{x^2+y^2}|2x+y| \leqslant |2x+y| $$

[lethal fossil]

aL

[runic halo]

can you make |2x+y|< eps?

@unkempt pine

[unkempt pine]

uhh i can't find a way tbh

@runic halo

[unkempt pine]

i meant the term to set < epsilon

[runic halo]

lose the "the"

let's try this

$$ |2x+y| \leqslant 2|x| + |y| < 2\frac{\varepsilon}{3} + \frac{\varepsilon}{3} = \varepsilon $$

[lethal fossil]

aL

[runic halo]

so if we assume |x|,|y| < eps /3, what restriction does that set to x^2+y^2 ?

pick delta = eps / 3 and see where that takes you @unkempt pine

[unkempt pine]

oke but what about delta then? @runic halo