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Evaluate the limits or explain why the limit fails to exist.

lim(x,y)(0,0)(x+y)2x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { ( x + y ) ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }


Step 1
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To prove that the limit does not exist we can rearrange the given terms using the properties of limit.


The first term is obviously equal to 11. For the second term take the paths x=tx=t and y=2ty=2t and let t=0t=0.

We now have:

limt04t25t2=45\lim_{t\to 0}\dfrac{4t^2}{5t^2}=\dfrac{4}{5}

However if we set x=tx=t and take y=3ty=3t we get a different result:

limt06t210t2=35\lim_{t\to 0}\dfrac{6t^2}{10t^2}=\dfrac{3}{5}

Therefore limit does not exist.

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