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

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

Question

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

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

Quizplus · Accepted Answer

DThe limit does not exist because approaching (0,0) along different paths yields different results. For example, approaching along the line y=x gives a limit of 1, while approaching along the line y=2x gives a limit of 4/5. However, if we approach along any line x=0 or y=0, the expression simplifies to 0, indicating that the limit depends on the path taken, hence it does not exist.

The Limit lim⁡(x,y)→(0,0)4xy2x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 4 x y } { 2 x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​2x2+y24xy​

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