examine the behavior of $f(x,y)=\frac{3x^{2}y}{7x^{4}+y^{2}}$ as $(x,y)$ approaches $(0,0)$. use \infinity\…

examine the behavior of $f(x,y)=\frac{3x^{2}y}{7x^{4}+y^{2}}$ as $(x,y)$ approaches $(0,0)$. use \infinity\ for \∞\ and \-infinity\ for \-∞\. use \dne\ for \does not exist\. (a) taking a linear approach to the origin along the $y$-axis, we find $lim_{(x,y)\to(0,0),x = 0}(\frac{3x^{2}y}{7x^{4}+y^{2}})=$ dne. (b) taking a linear approach to the origin along the $x$-axis, we find $lim_{(x,y)\to(0,0),y = 0}(\frac{3x^{2}y}{7x^{4}+y^{2}})=$. (c) taking a linear approach to the origin along a line $y=mx$ with $m\neq0$ we find $lim_{(x,y)\to(0,0),y = mx}(\frac{3x^{2}y}{7x^{4}+y^{2}})=lim_{x\to0}(\frac{3x^{3}m}{7x^{4}+(mx)^{2}})=$

examine the behavior of $f(x,y)=\frac{3x^{2}y}{7x^{4}+y^{2}}$ as $(x,y)$ approaches $(0,0)$. use \infinity\ for \∞\ and \-infinity\ for \-∞\. use \dne\ for \does not exist\. (a) taking a linear approach to the origin along the $y$-axis, we find $lim_{(x,y)\to(0,0),x = 0}(\frac{3x^{2}y}{7x^{4}+y^{2}})=$ dne. (b) taking a linear approach to the origin along the $x$-axis, we find $lim_{(x,y)\to(0,0),y = 0}(\frac{3x^{2}y}{7x^{4}+y^{2}})=$. (c) taking a linear approach to the origin along a line $y=mx$ with $m\neq0$ we find $lim_{(x,y)\to(0,0),y = mx}(\frac{3x^{2}y}{7x^{4}+y^{2}})=lim_{x\to0}(\frac{3x^{3}m}{7x^{4}+(mx)^{2}})=$

Answer

Explanation:

Step1: Evaluate limit along y - axis (x = 0)

When (x = 0), the function (f(x,y)=\frac{3x^{2}y}{7x^{4}+y^{2}}) becomes (\frac{3\times0^{2}\times y}{7\times0^{4}+y^{2}} = 0). So (\lim_{(x,y)\to(0,0),x = 0}\frac{3x^{2}y}{7x^{4}+y^{2}}=0).

Step2: Evaluate limit along x - axis (y = 0)

When (y = 0), the function (f(x,y)=\frac{3x^{2}y}{7x^{4}+y^{2}}) becomes (\frac{3x^{2}\times0}{7x^{4}+0^{2}} = 0). So (\lim_{(x,y)\to(0,0),y = 0}\frac{3x^{2}y}{7x^{4}+y^{2}}=0).

Step3: Evaluate limit along y=mx

Substitute (y = mx) into the function (f(x,y)): [ \begin{align*} \lim_{x\to0}\frac{3x^{2}(mx)}{7x^{4}+(mx)^{2}}&=\lim_{x\to0}\frac{3mx^{3}}{7x^{4}+m^{2}x^{2}}\ &=\lim_{x\to0}\frac{3mx^{3}}{x^{2}(7x^{2}+m^{2})}\ &=\lim_{x\to0}\frac{3mx}{7x^{2}+m^{2}}\ & = 0 \end{align*} ]

Answer:

Since the limit of (f(x,y)=\frac{3x^{2}y}{7x^{4}+y^{2}}) as ((x,y)) approaches ((0,0)) along the (y - axis) ((x = 0)), (x - axis) ((y = 0)) and the line (y=mx) is (0), (\lim_{(x,y)\to(0,0)}\frac{3x^{2}y}{7x^{4}+y^{2}} = 0)