\[ \begin{array}{l} \lim _{(x, y) \rightarrow(0,0)} \frac{8 x^{2} y^{2}}{x^{2}+y^{4}} \\ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3}+x y^{2}}{x^{2}+y^{2}} \\ \lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}} \\ \lim _{(x, y) \rightarrow(0,0)} \frac{x y+1}{x^{2}+y^{2}+1} \end{array} \] IREANDINQ dación Universitaria del Área Andin Actividad 1_Calculo Multivariable

To tackle limits involving multivariable functions, we often explore different paths to ensure the limit truly exists. For instance, let's start with the limit \(\lim _{(x, y) \rightarrow(0,0)} \frac{8 x^{2} y^{2}}{x^{2}+y^{4}}\). If we approach along the line \(y = mx\) (where \(m\) is a constant), we can see how the behavior changes based on the ratio between \(x\) and \(y\). By substituting \(y = mx\), we can simplify and find the limit as \(x\) approaches 0. Next, consider the limit \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}\). Using polar coordinates, where \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), can provide insights. As \(r\) approaches 0, the behavior of the limit becomes clearer because it transforms the expression into a function of \(\theta\), revealing if the limit depends on the direction or value of \(\theta\).