Question

HElP Me PLEASE
Given f of x is equal to the quantity 4x minus 7 end quantity divided by the quantity 8x plus 8 end quantity, what is the end behavior of the function?

As x → -∞, f(x) → 0.5; as x → ∞, f(x) → 0.5.
As x → -∞, f(x) → -0.5; as x → ∞, f(x) → -0.5.
As x → -∞, f(x) → 0.73; as x → ∞, f(x) → 0.73.
As x → -∞, f(x) → -0.73; as x → ∞, f(x) → -0.73.

Answer 1

Answer:

As x → -∞, f(x) → 0.5; as x → ∞, f(x) → 0.5

Step-by-step explanation:

Given function:

$f(x)=\dfrac{4x-7}{8x+8}$

Asymptote: a line that the curve gets infinitely close to, but never touches.

As the degrees of the numerator and denominator of the given function are equal, there is a horizontal asymptote at  $y=\dfrac{a}{b}$  (where a is the leading coefficient of the numerator, and b is the leading coefficient of the denominator).  This is the end behavior.

$\textsf{Horizontal asymptote}:y=\dfrac{4}{8}=\dfrac{1}{2}$

This is because as $x \rightarrow \infty$ the -7 of the numerator and the +8 of the denominator become negligible.  Therefore, we are left with:

$f(x) \rightarrow \dfrac{4x}{8x}$

Therefore:

$\textsf{As }\:x \rightarrow - \infty, f(x) \rightarrow 0.5$

$\textsf{As }\:x \rightarrow \infty, f(x) \rightarrow 0.5$