Use polar coordinates and L'Hopital's Rule to find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 10 \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }$
A) $\frac { \sqrt { 3 } } { 2 }$
B) 10
C) $\frac { 1 } { 2 }$
D) $\frac { \sqrt { 2 } } { 2 }$
E) 0

Question

Quizplus · Accepted Answer

In polar coordinates, $x^2 + y^2 = r^2$. Substituting this into the given limit, we get $\lim _ { r \rightarrow 0 } \frac { 10 \sin ( r^2 ) } { r^2 }$. As $r \rightarrow 0$, this limit approaches $10$, because $\lim _ { u \rightarrow 0 } \frac { \sin u } { u } = 1$ where $u = r^2$.

Use Polar Coordinates and L'Hopital's Rule to Find the Limit lim⁡(x,y)→(0,0)10sin⁡(x2+y2)x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 10 \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y210sin(x2+y2)​

Use Polar Coordinates and L'Hopital's Rule to Find the Limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 10 \sin \left( x ^ { 2 } + y ^ { 2 } \right) } { x ^ { 2 } + y ^ { 2 } }$