Question

Which of the following functions approach infinity as \( x \) approaches infinity?

A) \( f(x) = -\frac{1}{3}(x^3)(2x^5)(3x-9) \)

B) \( f(x) = -\frac{3}{4}(3x-1)(3x-1)(x-4)(x+7) \)

C) \( f(x) = (x+9)(x+7)(x+3)(x-1) \)

D) \( f(x) = 3x(2x-7)(x+5) \)

E) \( f(x) = -\frac{25}{x}(4x-1)(3x+8)(x-2)(x-5) \)

F) \( f(x) = 12(x-4)(x-8)(x-11) \)

Answer 1

`D) \( f(x) = 3x(2x-7)(x+5) \)` The function's dominant term
`\( 6x^3 \)`plays a pivotal role in showcasing the growth of the function without bound, confirming that as
`\( x \)`approaches infinity, f(x also approaches infinity due to the increasing cubic term. Consequently, the function
`\( f(x) = 3x(2x-7)(x+5) \)` satisfies the condition of approaching infinity as x approaches infinity.

Explanation:

The function
`\( f(x) = 3x(2x-7)(x+5) \)` approaches infinity as
`\( x \)`approaches infinity because it contains terms with the highest powers of x being cubic. When simplified, the dominant term becomes
`\( 6x^3 \),` indicating that the function's behavior grows without bound as
`\( x \)` increases without limit. The leading term
`\( 6x^3 \)`demonstrates the function's tendency to increase indefinitely asx moves toward infinity, affirming that this function's values approach infinity under these conditions.

This behavior is primarily dictated by the term with the highest power of x (cubic in this case), leading to an unbounded increase as x moves toward infinity. The rest of the terms either have lower powers or are constants, which do not significantly impact the overall behavior of the function as x becomes larger.

The function's dominant term
`\( 6x^3 \)`plays a pivotal role in showcasing the growth of the function without bound, confirming that as
`\( x \)`approaches infinity, f(x also approaches infinity due to the increasing cubic term. Consequently, the function
`\( f(x) = 3x(2x-7)(x+5) \)` satisfies the condition of approaching infinity as x approaches infinity.