Question

a rational function is displayed.
f(x)=\frac{(3 - 8x)}{(4x + 1)}
which of the following statements describes the end behavior of this function?
as ( x
ightarrow-infty,f(x)
ightarrow2 ).
as ( x
ightarrow-infty,f(x)
ightarrow-2 ).
as ( x
ightarrowinfty,f(x)
ightarrow\frac{3}{4} ).
as ( x
ightarrowinfty,f(x)
ightarrow-\frac{1}{4} ).

Explanation:

Step1: Analyze the degrees of numerator and denominator

The degree of the numerator \(3 - 8x\) (degree \(n = 1\)) and the degree of the denominator \(4x+1\) (degree \(m = 1\)).

Step2: Use the rule for horizontal asymptote when \(n=m\)

For a rational function \(y=\frac{a_nx^n+\cdots}{b_mx^m+\cdots}\), when \(n = m\), \(y=\frac{a_n}{b_m}\). Here \(a_n=-8\), \(b_m = 4\).

Step3: Calculate the limit as \(x\to\pm\infty\)

\(\lim_{x\to\pm\infty}f(x)=\lim_{x\to\pm\infty}\frac{3 - 8x}{4x + 1}=\lim_{x\to\pm\infty}\frac{\frac{3}{x}-8}{4+\frac{1}{x}}\) (divide numerator and denominator by \(x\)).
As \(x\to\pm\infty\), \(\frac{3}{x}\to0\) and \(\frac{1}{x}\to0\). So \(\lim_{x\to\pm\infty}f(x)=\frac{- 8}{4}=-2\).

Answer:

As \(x\to-\infty\), \(f(x)\to - 2\)