Question

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Answer 1

$\lim_{\ (x,y)\to\ (1,0)}\frac{y(x-1)}{(x-1)^2+y^2} \\ \\ \lim_{\ (x,y)\to\ (1,0)}\frac{y}{(x-1)+y^2}$

Fixando x = 1, temos:

$\lim_{\ (x=1,y)\to\ 0}\frac{y}{(1-1)+y^2} \\ \\ \lim_{\ (x,y)\to\ (1,0)}\frac{y}{y^2} \\ \\ \lim_{\ (x,y)\to\ (1,0)}\frac{1}{y}$

Limite tende ao infinito

Fixando y = 0

$\lim_{\ (x,y=0)\to\ 1}\frac{0}{(x-1)+0^2} \\ \\ \lim_{\ (x,y)\to\ (1,0)}\frac{0}{(x-1)}$

Limite tende a zero

Logo o limite não existe