Graphs

[Andy005]

For the graph x^3+y^3=1, why is there an asymptote at y=-x?

Thanks.

[Paradoxica]

Consider the graph when x and y are very very large. (x+y)(x²-xy+y²)=1 ⇒ x+y = 1/(x²-xy+y²) ≈ 0 ⇒ x≈-y

[fan96]

When ,



Which has no solution.

[mrbunton]

making y the subject:
y = (-x^3+1)^1/3
factor out x^3
=(x^3)^1/3 * (-1+1/x^3)^1/3
=x (-1+1/x^3)^1/3
as x becomes larger; the part with 1/x^3 becomes smaller and the graph of y=x (-1 + 1/x^3 )^1/3 gets closer and closer to the graph y=x(-1+0)^1/3 or y=-x. of course there will be no solutions but that is just because it's an asymtote and those never have solutions (take for example y=1/x which has an asymtote at y=0; yet one cannot arrive at a solution; same thing with y=e^x)