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

Question

Quizplus · Accepted Answer

The limit of $\frac{\sin(z)}{z}$ as $z$ approaches 0 is a well-known limit equal to 1. In this case, $z = x^2 + y^2$, which also approaches 0 as $(x, y) \rightarrow (0,0)$. Therefore, the limit of the given function as $(x, y) \rightarrow (0,0)$ is 1.

Evaluate lim⁡(x,y)→(0,0)sin⁡(x2+y2)x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y2sin(x2+y2)​

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