Question

Identify the vertical and horizontal asymptotes for f(x) = (2*2) + (4 / x^2) + (5x+6) .

A) Vertical asymptotes: x=−2, x=−3; Horizontal asymptote: y=0
B) Vertical asymptotes: x=−2, x=3; Horizontal asymptote: y=0
C) Vertical asymptotes: x=2, x=3; Horizontal asymptote: y=0
D) Vertical asymptotes: x=−3, x=−2; Horizontal asymptote: y=0.

Answer 1

The function
`\( f(x) = (2 \cdot 2) + \left( (4)/(x^2) \right) + (5x + 6) \)` has a vertical asymptote at ( x = 0 and no horizontal asymptote.. None of the options are correct.

Step-by-step explanation:

To find the vertical and horizontal asymptotes for the function
`\( f(x) = (2 \cdot 2) + \left( (4)/(x^2) \right) + (5x + 6) \)`, we need to analyze the behavior of the function as x approaches certain critical values.

First, let's simplify the function:

`\[ f(x) = 4 + (4)/(x^2) + 5x + 6 \]`

Now let's look for vertical asymptotes.

Vertical asymptotes occur where the function is undefined, which usually corresponds to zeros of the denominator in any fraction within the function.

The only fraction in f(x) is
`\( (4)/(x^2) \).`

This expression is undefined when
`\( x^2 = 0 \)`, which would imply ( x = 0).

Therefore, there is a vertical asymptote at ( x = 0).

Next, we will find the horizontal asymptotes by analyzing the behavior of f(x) as x approaches
`\( \infty \)` and
`\( -\infty \)`.

For large values of x, the term
`\( (4)/(x^2) \)` approaches 0, and the dominant term in the function becomes 5x, since the degree of x in this term is higher than for any other term in the function.

Thus, as x approaches
`\( \infty \)` or
`\( -\infty \)`, the value of
`\( f(x) \)` is determined by
`\( 5x \),` which increases without bound.

This means there is no horizontal asymptote in the traditional sense (where the function approaches a constant value as x goes to
`\( \infty \)` or
`\( -\infty \))` for this function.

Based on this analysis, option A, B, C, and D all incorrectly claim a horizontal asymptote at ( y = 0), which is not the case for this function.

Additionally, none of the given vertical asymptotes match the one found here at \( x = 0 \).

Since none of the options correctly state the asymptotes of the function f(x), the proper action would be to indicate that the asymptotes do not match any of the given options. It's possible there's been a mix-up in the creation of the question or the answer choices provided.

The correct answer should state that ( f(x)) has a vertical asymptote at ( x = 0 and no horizontal asymptote.