The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { y ^ { 4 } + 3 x ^ { 2 } y ^ { 2 } + 2 x y ^ { 3 } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } }$ is

A)Does not exist
B)2
C)4
D)0
E)1

Question

The limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { y ^ { 4 } + 3 x ^ { 2 } y ^ { 2 } + 2 x y ^ { 3 } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } }$ is

A)Does not exist
B)2
C)4
D)0
E)1

Quizplus · Accepted Answer

The limit does not exist. To show this, consider approaching (0,0) along different paths. For example, if you approach along the line y=x, the expression simplifies differently than if you approach along the line y=mx (where m is a constant different from 1), leading to different limit values. This inconsistency implies the limit does not exist.

The Limit lim⁡(x,y)→(0,0)y4+3x2y2+2xy3(x2+y2)2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { y ^ { 4 } + 3 x ^ { 2 } y ^ { 2 } + 2 x y ^ { 3 } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } }lim(x,y)→(0,0)​(x2+y2)2y4+3x2y2+2xy3​

The Limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { y ^ { 4 } + 3 x ^ { 2 } y ^ { 2 } + 2 x y ^ { 3 } } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } }$