What is the limit as (x,y) approaches (0,0) of (x^{10}-y^{10})/(x^5+y^5) ?

The limit as (x,y) approaches (0,0) of (x^{10}-y^{10})/(x^5+y^5) is 0

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limit as (x,y) approaches (0,0) of (x^{10}-y^{10})/(x^5+y^5)

(x, y) \to (0, 0) lim (\frac{x ^{10} - y ^{10}}{x ^{5} + y ^{5}})

0

Solution steps

(x, y) \to (0, 0) lim (\frac{x ^{10} - y ^{10}}{x ^{5} + y ^{5}})

Simplify

\frac{x ^{10} - y ^{10}}{x ^{5} + y ^{5}} : (x - y) (x^{4} + x^{3} y + x^{2} y^{2} + x y^{3} + y^{4})

= (x, y) \to (0, 0) lim ((x - y) (x^{4} + x^{3} y + x^{2} y^{2} + x y^{3} + y^{4}))

Plug in the value

(x, y) = (0, 0)

= (0 - 0) (0^{4} + 0^{3} \cdot 0 + 0^{2} \cdot 0^{2} + 0 \cdot 0^{3} + 0^{4})

Simplify

(0 - 0) (0^{4} + 0^{3} \cdot 0 + 0^{2} \cdot 0^{2} + 0 \cdot 0^{3} + 0^{4}) : 0

= 0

y^{^{'''}} - 3 y^{^{''}} + 3 y^{^{'}} - y = x - 9 e^{x}

y^{^{'}} + 2 y = x

\int - 5 x^{2} d x

\frac{d}{d x} (\frac{( x ^{2} + 1 )}{x + 4})

\frac{d}{d x} ((x - 1)^{2} - (x^{2} + 1)^{4})

What is the limit as (x,y) approaches (0,0) of (x^{10}-y^{10})/(x^5+y^5) ?
The limit as (x,y) approaches (0,0) of (x^{10}-y^{10})/(x^5+y^5) is 0