Consider lim_(x, y) rightarrow (0, 0) 6x^3 y^4/x^5 + y^10 Set x = 0 and let y ri

Question

Consider lim_(x, y) rightarrow (0, 0) 6x^3 y^4/x^5 + y^10 Set x = 0 and let y rightarrow 0 (Approaching origin along the y-axis.) What is the limit along this path? What limit do you get when you approach the origin along an arbitrary straight line (y = mx)? What limit do you get when you approach the origin along a parabola y = x^2 ? What limit do you get when you approach the origin along a parabola x = y^2 ? Evaluate lim_(x, y) rightarrow (0, 0) 6x^3 y^4/x^5 + y^10. Determine if the following functions are continuous. f (x, y) = {(x^2 + y^2)^2 + 2 (x^2 + y^2)/x^2 + y^2 if (x, y) notequalto (0, 0) 2 if (x, y) = (0, 0)

Explanation / Answer

lt(x,y=0,0) = 6x3y4/x5+y10

at y = 0 then 0 for the eqation and also for x = 0 for equation

for , y = mx

then 6x3.(mx)4/x5+(mx)10

6mx7/x5+ m10x10

x5(6mx2)/x5(1+m10x5)

6mx2/(1+m10x5)

c)

x2=y

6(x2).x.y4/(x4.x+y10)

6y.y1/2.y4/(y2.y1/2+y10)

6y11/2/(y5/2+y10)

d) for x = 0 y = 0

for y = 0 then 0

for x = 0 then 0

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