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To prove this assertion, we can use the Squeeze Theorem, which states that if g(x) ≤ f(x) ≤ h(x) for every x in an interval, and if lim x → a g(x) = lim x → a h(x) = L for a point a in the interval, then lim x → a f(x) = L. g

Respuesta :

aliakbar921853

Answer:

Limit is equal to zero

Step-by-step explanation:

We have limit as

         0 ≤  [tex]\frac{xy}{x^{2} + y^{2} }[/tex] ≤  x

Now,

let g (x) = 0 (constant)              

    h (x,y = x)

Also, we have

   [tex]\lim_{(x,y) \to \((0,0)} 0 = 0[/tex]    

Also,  

[tex]\lim_{(x,y) \to \((0,0)} IxI = 0[/tex]

Now by using given theorem, we get

   [tex]\lim_{(x,y) \to \(0,0)} I\frac{xy}{\sqrt{x^{2}+y^{2} } }I[/tex]= 0

Hence proved that limit is equal to zero.

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