limit as x,y approaches (0,0) of ((2x^2y))/(x^2+y^4)

解

x, y \to (0, 0) lim (\frac{( 2 x ^{2} y )}{x ^{2} + y ^{4}})

0

解答ステップ

x, y \to (0, 0) lim (\frac{2 x ^{2} y}{x ^{2} + y ^{4}})

極座標に変換する

x = r cos (θ)

= r \to 0 lim (\frac{2 ( r cos ( θ ) ) ^{2} r sin ( θ )}{( r cos ( θ ) ) ^{2} + ( r sin ( θ ) ) ^{4}})

簡素化

\frac{2 ( r cos ( θ ) ) ^{2} r sin ( θ )}{( r cos ( θ ) ) ^{2} + ( r sin ( θ ) ) ^{4}} : \frac{2 r cos ^{2} ( θ ) sin ( θ )}{r ^{2} sin ^{4} ( θ ) + cos ^{2} ( θ )}

= r \to 0 lim (\frac{2 r cos ^{2} ( θ ) sin ( θ )}{r ^{2} sin ^{4} ( θ ) + cos ^{2} ( θ )})

値を挿入する

r = 0

= \frac{2 \cdot 0 \cdot cos ^{2} ( θ ) sin ( θ )}{0 ^{2} sin ^{4} ( θ ) + cos ^{2} ( θ )}

簡素化

\frac{2 \cdot 0 \cdot cos ^{2} ( θ ) sin ( θ )}{0 ^{2} sin ^{4} ( θ ) + cos ^{2} ( θ )} : 0

= 0