\( \lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { - x y ^ { 2 } } { x ^ { 2 } + y ^ { 4 } } \)

To analyze the limit \( \lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { - x y ^ { 2 } } { x ^ { 2 } + y ^ { 4 } } \), we can use different paths to see if the limit is consistent. 1. **Approach along the line y = 0**: \[ \frac{-x(0)^2}{x^2 + (0)^4} = 0. \] 2. **Approach along the line x = 0**: \[ \frac{0 \cdot y^2}{0 + y^4} = 0. \] 3. **Approach along the line y = kx (for some constant k)**: \[ \lim_{x \to 0} \frac{-x(kx)^2}{x^2 + (kx)^4} = \lim_{x \to 0} \frac{-kx^3}{x^2 + k^4x^4} = \lim_{x \to 0} \frac{-k x^3}{x^2(1 + k^4 x^2)}. \] This simplifies to: \[ \lim_{x \to 0} \frac{-k x}{1 + k^4 x^2} = 0. \] By testing various paths, we consistently find that the limit approaches 0. Thus, the final result is: \[ \lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { - x y ^ { 2 } } { x ^ { 2 } + y ^ { 4 } } = 0. \]