Find the horizontal and vertical asymptotes of the curve.
y = 4+x^4/x^2−x^4
x=? (smallest x-value)
x=?
x=? (largest x-value)
y=?
same stuff
(any reason why you switched names ?)
To find the horizontal and vertical asymptotes of the curve represented by the equation y = (4 + x^4) / (x^2 - x^4), we need to analyze the behavior of the function as x approaches positive or negative infinity.
1. Horizontal Asymptotes:
To determine the horizontal asymptotes, we look at the limit of the function as x approaches infinity and negative infinity.
For x approaching infinity, we consider the highest power of x in the numerator and denominator, which is x^4. Dividing each term by x^4, we get:
y = (4 / x^4 + 1)
As x approaches infinity, the term 4 / x^4 approaches zero, and the function simplifies to:
y ≈ 0 + 1 = 1
Therefore, y = 1 is the horizontal asymptote as x approaches infinity.
For x approaching negative infinity, we perform the same process:
y = (4 / x^4 + 1)
As x approaches negative infinity, the term 4 / x^4 also approaches zero, and the function simplifies to:
y ≈ 0 + 1 = 1
So, y = 1 is the horizontal asymptote as x approaches negative infinity as well.
2. Vertical Asymptotes:
To find the vertical asymptotes, we need to check the values that make the denominator equal to zero by solving the equation x^2 - x^4 = 0.
Factorizing the equation gives:
x^2(1 - x^2) = 0
This equation has two solutions: x = 0 and x = ±1.
Therefore, x = 0, x = 1, and x = -1 are the vertical asymptotes.
Now, let's find the smallest and largest x-values and the corresponding y-values.
The function is defined for all x except when x = 0, x = 1, and x = -1 due to the vertical asymptotes.
Therefore, the smallest x-value is the smallest real number excluding the vertical asymptotes, which can be a large negative number.
The largest x-value is the largest real number excluding the vertical asymptotes, which can be a large positive number.
To find the smallest and largest y-values, substitute the smallest and largest x-values into the equation and evaluate the function.
Unfortunately, since we don't have specific numerical values or ranges for x, we cannot provide the exact x-values or the corresponding y-values.