Find the horizontal asymptote of the graph of y = Start Fraction negative 2 x superscript 6 baseline plus 5 x plus 8 over 8 x superscript 6 baseline plus 6 x plus 5 End Fraction
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y = (-2x^6 + 5x + 8)/(8x^6 + 6x + 5)
since the numerator and denominator are the same degree, the horizontal asymptote is the line y = -2/8
To find the horizontal asymptote of the graph of y = (−2x^6 + 5x + 8)/(8x^6 + 6x + 5), you need to compare the degrees of the numerator and the denominator.
The degree of the numerator is 6, and the degree of the denominator is also 6. When the degrees are the same, the horizontal asymptote can be determined by dividing the leading coefficients of the numerator and the denominator.
In this case, the leading coefficient of the numerator is -2, and the leading coefficient of the denominator is 8. Therefore, the horizontal asymptote is the horizontal line given by the equation y = -2/8, which simplifies to y = -1/4.
So, the horizontal asymptote of the graph of y = (-2x^6 + 5x + 8)/(8x^6 + 6x + 5) is y = -1/4.
To find the horizontal asymptote of the graph of the given function, you need to examine the degrees of the numerator and denominator polynomials.
The given function is y = (−2x^6 + 5x + 8) / (8x^6 + 6x + 5).
1. Begin by comparing the degrees of the numerator and denominator polynomials. The highest degree term of the numerator is -2x^6, and the highest degree term of the denominator is 8x^6. Since the degrees are the same, we can focus on comparing the coefficients of the highest degree terms.
2. Divide both the numerator and denominator by x^6 to simplify the expression:
y = (-2 + (5/x^5) + (8/x^6)) / (8 + (6/x^5) + (5/x^6))
3. As x approaches positive or negative infinity, the terms involving 1/x^5 and 1/x^6 become very small compared to the other terms. The coefficients of these terms, 5/x^5 and 8/x^6, approach 0 as x approaches infinity.
4. Therefore, when x approaches positive or negative infinity, the simplified expression becomes:
y ≈ (-2) / (8)
y ≈ -2/8
y ≈ -1/4
5. This means that the horizontal asymptote of the given function is y = -1/4.
To summarize, the horizontal asymptote of the graph of y = (-2x^6 + 5x + 8) / (8x^6 + 6x + 5) is y = -1/4.
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To find the horizontal asymptote of this function, we need to look at the highest degree terms in the numerator and denominator. Both the numerator and denominator have terms with degree 6, so we divide the coefficient of the highest degree term in the numerator (-2) by the coefficient of the highest degree term in the denominator (8). This gives us:
y = (-2/8) x^6 + 5x + 8/(8x^6 + 6x + 5)
Simplifying this, we get:
y = (-1/4) x^6 + 5x + (8/(8x^6 + 6x + 5))
As x gets very large (either positively or negatively), the x^6 term in the denominator will dominate the other terms, since its exponent is so much larger than the exponents on the other terms. This means that the denominator will grow much faster than the numerator, and the fraction as a whole will approach 0. Therefore, the horizontal asymptote of the function is y = 0.