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

examine the behavior of (f(x,y)=\frac{3x^{2}y}{7x^{4}+y^{2}}) as ((x,y)) approaches ((0,0)). (a) taking a linear approach to the origin along the (y -)axis, we find (lim_{(x,y)\to(0,0),x = 0}left(\frac{3x^{2}y}{7x^{4}+y^{2}}\right)=). (b) taking a linear approach to the origin along the (x -)axis, we find (lim_{(x,y)\to(0,0),y = 0}left(\frac{3x^{2}y}{7x^{4}+y^{2}}\right)=). (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}left(\frac{3x^{2}y}{7x^{4}+y^{2}}\right)=lim_{x\to0}left(\right)=). thought question: do your answers to parts (a)-(c) allow you to conclude that the limit exists?

examine the behavior of (f(x,y)=\frac{3x^{2}y}{7x^{4}+y^{2}}) as ((x,y)) approaches ((0,0)). (a) taking a linear approach to the origin along the (y -)axis, we find (lim_{(x,y)\to(0,0),x = 0}left(\frac{3x^{2}y}{7x^{4}+y^{2}}\right)=). (b) taking a linear approach to the origin along the (x -)axis, we find (lim_{(x,y)\to(0,0),y = 0}left(\frac{3x^{2}y}{7x^{4}+y^{2}}\right)=). (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}left(\frac{3x^{2}y}{7x^{4}+y^{2}}\right)=lim_{x\to0}left(\right)=). thought question: do your answers to parts (a)-(c) allow you to conclude that the limit exists?

Answer

Explanation:

Step1: Find limit along y - axis (a)

Set (x = 0). Then (f(x,y)=\frac{3x^{2}y}{7x^{4}+y^{2}}) becomes (\lim_{(x,y)\to(0,0),x = 0}\frac{3x^{2}y}{7x^{4}+y^{2}}=\lim_{y\to0}\frac{0}{y^{2}} = 0).

Step2: Find limit along x - axis (b)

Set (y = 0). Then (f(x,y)=\frac{3x^{2}y}{7x^{4}+y^{2}}) becomes (\lim_{(x,y)\to(0,0),y = 0}\frac{3x^{2}y}{7x^{4}+y^{2}}=\lim_{x\to0}\frac{0}{7x^{4}} = 0).

Step3: Find limit along (y=mx) (c)

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

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

Answer:

The limit exists and is equal to (0).