5-) \( \lim _{(x, y) \rightarrow(0,0)} \frac{x-y+2 \sqrt{x}-2 \sqrt{y}}{\sqrt{x}-\sqrt{y}} \) 6-) \( \lim _{(x, y) \rightarrow(2,0)} \frac{\sqrt{2 x-y}-2}{2 x-y-4} \) 7-) \( \lim _{(x, y) \rightarrow(2,-4))^{2} y-x y+4 x^{2}-4 x} \) 8-) \( \lim _{(x, y) \rightarrow(0,0)} \frac{4 x y}{3 y^{2}-x^{2}} \) 9-) \( \lim _{(x, y) \rightarrow(1,2)} \frac{x y-2 x-y+2}{x^{2}-2 x+y^{2}-4 y+5} \) 10-) \( \lim _{(x, y) \rightarrow(0,0)} \frac{\frac{3}{x} y^{2}}{x+y^{3}} \) 11-) \( \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y-x^{2}-y^{2}}{x^{2}+y^{2}} \) 12-) \( \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3}+4 x^{2}+2 y^{2}}{2 x^{2}+y^{2}} \)

Oh, limits! They can be quite the puzzle, can't they? When dealing with limits in multiple variables, it's essential to remember that the path you choose to approach the point can influence the outcome. Always try testing different paths; you might be surprised by how different results can lead to similar conclusions! When tackling these limits, one common mistake is overlooking the importance of simplifying the expression before substituting values. Factorization or even rationalizing can reveal behaviors that direct substitution can obscure. Keep your eyes peeled for indeterminate forms and consider methods like polar coordinates for tricky limits—turning those messy Cartesian coordinates into a sleeker, circular form can sometimes unveil the path to the limit's answer!