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
To find the horizontal asymptote of the given function, we need to determine the limit of the function as x approaches infinity.
We first simplify the expression by dividing all terms by the highest power of x in the denominator, which is x^6:
y = (-2x^6 + 5x + 8) / (8x^6 + 6x + 5)
y = (-2 + 5/x^5 + 8/x^6) / (8 + 6/x^5 + 5/x^6)
As x approaches infinity, the terms with 1/x or 1/x^2 will approach zero, leaving us with:
y = (-2 + 0 + 0) / (8 + 0 + 0)
y = -2 / 8
y = -1/4
Therefore, the horizontal asymptote of the graph of the function y = (-2x^6 + 5x + 8) / (8x^6 + 6x + 5) is y = -1/4.