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

A)Does not exist
B)2
C) $\frac { 1 } { 2 }$
D)0
E)1

Question

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

A)Does not exist
B)2
C) $\frac { 1 } { 2 }$
D)0
E)1

Quizplus · Accepted Answer

The 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 0, while approaching along the line y=0 gives a limit of 1.

The Limit lim⁡(x,y)→(0,0)x2−y2x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y2x2−y2​

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