Find the limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( 6 x ^ { 2 } + 6 y ^ { 2 } \right) } { 7 x ^ { 2 } + 7 y ^ { 2 } }$. Hint: If $x = r \cos \theta$ and $y = r \sin \theta$, then $( x , y ) \rightarrow ( 0,0 )$ if and only if $r \rightarrow 0 ^ { + }$.
Select the correct answer.
A) 1
B) 0
C) $\frac { 7 } { 6 }$
D) $\frac { 6 } { 7 }$

Question

Quizplus · Accepted Answer

Using polar coordinates, the expression becomes $\frac{\sin(6r^2)}{7r^2}$. As $r \to 0$, $\sin(6r^2) \approx 6r^2$, so the limit is $\frac{6r^2}{7r^2} = \frac{6}{7}$.

Find the Limit lim⁡(x,y)→(0,0)sin⁡(6x2+6y2)7x2+7y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( 6 x ^ { 2 } + 6 y ^ { 2 } \right) } { 7 x ^ { 2 } + 7 y ^ { 2 } }lim(x,y)→(0,0)​7x2+7y2sin(6x2+6y2)​

Find the Limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \sin \left( 6 x ^ { 2 } + 6 y ^ { 2 } \right) } { 7 x ^ { 2 } + 7 y ^ { 2 } }$