Find the asymptotes of x^2(x^2+2)=y^3(x+5)

1 answer

To find the asymptotes of the given equation, we need to analyze the behavior of the equation as x approaches positive or negative infinity.

First, let's rearrange the equation:

x^2(x^2+2) = y^3(x+5)

x^4 + 2x^2 = y^3(x+5)

Now, let's examine the behavior of the equation as x approaches positive or negative infinity:

As x approaches positive infinity:
When x becomes very large, the terms x^4 and 2x^2 will dominate, and the term y^3(x+5) will become insignificant. Therefore, we can neglect the y^3(x+5) term. The equation simplifies to:

x^4 + 2x^2 = 0

Solving this equation, we find that x = 0 is a possible asymptote as x approaches positive infinity.

As x approaches negative infinity:
Similarly, when x becomes very large in the negative direction, the terms x^4 and 2x^2 will dominate, and the term y^3(x+5) will become insignificant. Again, neglecting the y^3(x+5) term, the equation simplifies to:

x^4 + 2x^2 = 0

Solving this equation, we find that x = 0 is a possible asymptote as x approaches negative infinity as well.

Therefore, the equation has x = 0 as a possible asymptote.