Question

find all horizontal asymptotes of the following function.
$f(x) = \frac{x - 1}{(2x - 3)(x - 1)}$
answer attempt 5 out of 5
one horizontal asymptote
$\frac{x}{-3x}$

Explanation:

Step1: Simplify the function

First, we can cancel out the common factor \((x - 1)\) in the numerator and the denominator, but we need to note that \(x
eq1\) (because when \(x = 1\), the original function is undefined). So, \(f(x)=\frac{x - 1}{(2x - 3)(x - 1)}=\frac{1}{2x - 3}\) for \(x
eq1\).

Step2: Analyze the degrees of numerator and denominator

For a rational function \(y=\frac{N(x)}{D(x)}\), to find the horizontal asymptote, we compare the degrees of the numerator \(N(x)\) and the denominator \(D(x)\).

• The numerator of the simplified function \(f(x)=\frac{1}{2x - 3}\) is a constant (degree \(0\)), and the denominator is a linear function (degree \(1\)).
• When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\) (because as \(x

ightarrow\pm\infty\), the value of the function approaches \(0\)).

Answer:

The horizontal asymptote is \(y = 0\)