Evaluate the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 5 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$
A) the limit does not exist
B) $5$
C) 2
D) 0
E) 1

Question

Quizplus · Accepted Answer

We can approach the origin along the $x$-axis by setting $y=0$ in the expression to get $\frac{0}{x^2} = 0$. We can approach the origin along the line $y=mx$, where $m$ is any constant, to get 
$$\lim\limits_{(x,y)\to (0,0)} \frac{5xy^2}{x^2+y^2} = \lim\limits_{x\to 0} \frac{5mx^3}{x^2+m^2x^2} = \lim\limits_{x\to 0} \frac{5mx}{1+m^2} = 0$$
Therefore, the limit is $\boxed{\textbf{(D)}\ 0}$.

Evaluate the Limit lim⁡(x,y)→(0,0)5xy2x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 5 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }lim(x,y)→(0,0)​x2+y25xy2​

Evaluate the Limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 5 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$