Consider g (x) = StartFraction 4 x + 9 Over x Superscript 6 Baseline + 1 EndFraction Which statement correctly uses limits to determine the end behavior of g(x)? Limit of StartFraction 4 x + 9 Over x Superscript 6 Baseline + 1 EndFraction as x approaches plus-or-minus infinity = limit of StartFraction 4 Over 1 EndFraction as x approaches plus-or-minus infinity, so as x approaches infinity, g (x) approaches 4. Limit of StartFraction 4 x + 9 Over x Superscript 6 Baseline + 1 EndFraction 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 4. Limit of StartFraction 4 x + 9 Over x Superscript 6 Baseline + 1 EndFraction 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. Limit of StartFraction 4 x + 9 Over x Superscript 6 Baseline + 1 EndFraction as x approaches plus-or-minus infinity = limit of StartFraction 4 x Over 1 EndFraction as x approaches plus-or-minus infinity, so as x approaches infinity, g (x) approaches infinity.

Answer: C

Step-by-step explanation:

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.

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