What is the limit as (x,y) approaches (0,0) of ((xy^4))/(x^4+y^4) ?

The limit as (x,y) approaches (0,0) of ((xy^4))/(x^4+y^4) is 0

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

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

0

Solution steps

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

Convert to polar coordinates

x = r cos (θ)

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

Simplify

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

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

Plug in the value

r = 0

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

Simplify

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

= 0

What is the limit as (x,y) approaches (0,0) of ((xy^4))/(x^4+y^4) ?
The limit as (x,y) approaches (0,0) of ((xy^4))/(x^4+y^4) is 0