The horizontal asymptotes of the curve
y=12x/(x4+1)^(1/4)
As x-> infinity, y -> 12
That is a horizontal asymptote. I don't see any others.
Also, as x→-∞,
y→-12.
Thanks mathmate! I should have seen that
To find the horizontal asymptotes of a curve, we need to consider the behavior of the function as x approaches positive or negative infinity.
In this case, the given function is:
y = 12x / (x^4 + 1)^(1/4)
As x approaches positive or negative infinity, terms with smaller powers become negligible compared to terms with larger powers. Hence, we can simplify the expression as follows:
As x approaches positive infinity, x^4 becomes much larger than 1. Hence, we can ignore the term 1 in the denominator:
y ≈ 12x / (x^4)^(1/4)
= 12x / x
= 12
So, as x approaches positive infinity, y approaches 12.
As x approaches negative infinity, again, x^4 becomes much larger than 1. Hence, we can ignore the term 1 in the denominator:
y ≈ 12x / (x^4)^(1/4)
= 12x / x
= 12
So, as x approaches negative infinity, y approaches 12.
Therefore, the horizontal asymptotes of the curve y = 12x / (x^4 + 1)^(1/4) are y = 12.