Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { x ^ { 2 } + x y + y ^ { 2 } }$
A)0
B)1
C) $\frac { 1 } { 2 }$
D) $\frac { 1 } { 3 }$
E) $\frac { 1 } { 4 }$
F) $\frac { 1 } { 5 }$
G) $\frac { 1 } { 6 }$
H)Does not exist

Question

Evaluate $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { x ^ { 2 } + x y + y ^ { 2 } }$
A)0
B)1
C) $\frac { 1 } { 2 }$ 
D) $\frac { 1 } { 3 }$ 
E) $\frac { 1 } { 4 }$ 
F) $\frac { 1 } { 5 }$ 
G) $\frac { 1 } { 6 }$ 
H)Does not exist

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 to $\frac{x^2}{3x^2} = \frac{1}{3}$. However, if you approach along the line $y = -x$, the expression simplifies to $\frac{-x^2}{x^2} = -1$, which is not defined in the given expression, indicating a different behavior. These different results for different paths imply the limit does not exist.

Evaluate lim⁡(x,y)→(0,0)xyx2+xy+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x y } { x ^ { 2 } + x y + y ^ { 2 } }lim(x,y)→(0,0)​x2+xy+y2xy​

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