Find the horizontal asymptote, if any, of the graph of the given function.
g(x) = 4x^5+9x^4+1/9x^5+3x^4-9x^2+3x+1
To find the horizontal asymptote of a function, we need to examine the behavior of the function as x approaches positive infinity and negative infinity.
For the given function g(x) = (4x^5 + 9x^4 + 1) / (9x^5 + 3x^4 - 9x^2 + 3x + 1), we can determine the horizontal asymptote by comparing the degrees of the numerator and denominator polynomials.
The degree of the numerator is 5 (highest power of x), and the degree of the denominator is also 5.
When the degrees of the polynomials are equal, we can find the horizontal asymptote by comparing the coefficients of the terms with the highest power of x.
In this case, the coefficient of x^5 in both the numerator and denominator is 4.
Therefore, the horizontal asymptote of the graph of g(x) is y = 4.
To confirm this, we can take the limit of g(x) as x approaches positive and negative infinity.
As x approaches positive infinity:
lim(x->∞) (4x^5 + 9x^4 + 1) / (9x^5 + 3x^4 - 9x^2 + 3x + 1) = 4
As x approaches negative infinity:
lim(x->-∞) (4x^5 + 9x^4 + 1) / (9x^5 + 3x^4 - 9x^2 + 3x + 1) = 4
Both limits evaluate to 4, confirming that the horizontal asymptote is y = 4 for the graph of g(x).