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

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

Question

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

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

Quizplus · Accepted Answer

The limit approaches 0 because as $x$ and $y$ approach 0, the numerator $xy$ approaches 0 faster than the denominator $\sqrt{x^2 + y^2}$, leading to the whole expression approaching 0.

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

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