To find the end behavior of a function, we find it's limits as x approaches infinity, getting the correct option as:
As x approaches plus-or-minus infinity = limit of StartFraction 4 Over x Superscript 5 EndFraction as x approaches plus-or-minus infinity, so as x approaches infinity, g (x) approaches 0.
Function:
The function given is:
[tex]g(x) = \frac{4x+9}{x^6+1}[/tex]
Limit as x goes to infinity:
To find the limit of a function as x goes to infinity, we consider the term with the highest exponent in the numerator and in the denominator. So
[tex]\lim_{x \rightarrow \infty} g(x) = \lim_{x \rightarrow \infty} \frac{4x+9}{x^6+1} = \lim_{x \rightarrow \infty} \frac{4x}{x^6} = \lim_{x \rightarrow \infty} \frac{4}{x^5} = \frac{4}{\infty^5} = 0[/tex]
The graphic of the function, given at the end of this answer, corroborates the answer.
Thus, the correct option is:
As x approaches plus-or-minus infinity = limit of StartFraction 4 Over x Superscript 5 EndFraction as x approaches plus-or-minus infinity, so as x approaches infinity, g (x) approaches 0.
For more on limits as x approaches infinity, you can check brainly.com/question/12207599.
