Question

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 $m
eq0$ we find $lim_{(x,y)\to(0,0),y = mx}(\frac{3x^{2}y}{7x^{4}+y^{2}})=lim_{x\to0}(\frac{3x^{3}m}{7x^{4}+(mx)^{2}})=$

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 \(0\), \(\lim_{(x,y)\to(0,0)}\frac{3x^{2}y}{7x^{4}+y^{2}} = 0\)