Find the limit, if it exists, or show that the limit does not exist. lim(,)→(0,0) 2 2 4

A limit in mathematics is the value that a function approaches when its input approaches some value.
Limits are used to define continuity, derivatives, and integrals in calculus and mathematical analysis.

In order for such a limit to occur, the fraction [tex]\frac{x^{2} }{x^{2} +y^{2} }[/tex] must be comparable to the same value [tex]L[/tex], regardless of the way we take to get there [tex](0,0)[/tex].

Try approaching [tex](0,0)[/tex] along the x-axis.

This means setting [tex]y=0[/tex] and finding the limit [tex]lim_{x-0} \frac{x^{2} }{x^{2} +y^{2} }[/tex].

We obtain:

[tex]lim_{x-0,y=0}\frac{x^{2} }{x^{2} +y^{2} } =lim_{y=0}}\frac{x^{2} }{x^{2} +0 }\\=lim_{x-0}} \frac{x^{2} }{x^{2} } \\\\=lim_{x-0}}1\\=1[/tex]

Now evaluate approaching [tex](0,0)[/tex] along the y-axis.

This means setting [tex]x=0[/tex] and finding the limit [tex]lim_{y-0} \frac{x^{2} }{x^{2} +y^{2} }[/tex].

[tex]lim_{y-0,x-0} \frac{x^{2} }{x^{2} +y^{2} } =lim_{y-0} \frac{0}{0+y^{2} } \\=lim_{y-0} \frac{0}{y^{2} } \\=lim_{y-0} 0\\=0[/tex]

Approaching the origin via these two methods results in distinct limits.

[tex]lim_{x-0,y-0} \frac{x^{2} }{x^{2} +y^{2} }[/tex] ≠ [tex]lim_{y-0,x-0}\frac{x^{2} }{x^{2} +y^{2} }[/tex]

Therefore the limit does not exist.

Know more about limits here:

https://brainly.com/question/1521191

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The correct question is given below:

Find the limit, if it exists, or show that the limit does not exist.

[tex]lim_{(x,y) -(0,0)} \frac{x^{2} }{x^{2} +y^{2} }[/tex]