Evaluate the limit.
$$\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 7 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$$
A) 2
B) the limit does not exist
C) 7
D) 1
E) 0

Question

Quizplus · Accepted Answer

To evaluate the limit, we can approach along the line $y=mx$ for various values of $m$. So, we have
\begin{align*}
\lim_{(x,y)ightarrow(0,0)}\frac{7xy^2}{x^2+y^2}&=\lim_{xightarrow 0}\frac{7x(mx)^2}{x^2+(mx)^2}\
&=\lim_{xightarrow 0}\frac{7m^2x^4}{x^2(m^2+1)}\
&=\lim_{xightarrow 0}\frac{7m^2x^2}{m^2+1}=0.
\end{align*}
Since the limit along different paths is different, the limit does not exist. Thus, the best choice is $\boxed{	extbf{(B)}}$ the limit does not exist.

Evaluate the Limit lim⁡(x,y)→(0,0)7xy2x2+y2\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 7 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }(x,y)→(0,0)lim​x2+y27xy2​

Evaluate the Limit $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 7 x y ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$