What is the horizontal asymptote of [tex]$f(x)=\frac{-2 x}{x+1}$[/tex]?

A. [tex]y=-2[/tex]
B. [tex]y=-1[/tex]
C. [tex]y=0[/tex]
D. [tex]y=1[/tex]



Answer :

To determine the horizontal asymptote of the function [tex]\( f(x) = \frac{-2x}{x+1} \)[/tex], we need to analyze the behavior of the function as [tex]\( x \)[/tex] approaches positive and negative infinity.

### Step-by-Step Solution:

1. Identify the degrees of the numerator and the denominator:
- The numerator of [tex]\( f(x) \)[/tex] is [tex]\( -2x \)[/tex], which is a linear polynomial of degree 1.
- The denominator of [tex]\( f(x) \)[/tex] is [tex]\( x + 1 \)[/tex], which is also a linear polynomial of degree 1.

2. Compare the degrees:
- Both the numerator and the denominator are of the same degree (degree 1).

3. Determine the horizontal asymptote for rational functions:
- When the degrees of the numerator and denominator are the same, the horizontal asymptote is found by taking the ratio of the leading coefficients.
- The leading coefficient of the numerator [tex]\( -2x \)[/tex] is [tex]\(-2\)[/tex].
- The leading coefficient of the denominator [tex]\( x + 1 \)[/tex] is [tex]\(1\)[/tex].

4. Calculate the horizontal asymptote:
- The horizontal asymptote is given by the ratio [tex]\( \frac{-2}{1} \)[/tex].

Therefore, the horizontal asymptote for the function [tex]\( f(x) = \frac{-2x}{x+1} \)[/tex] is:

[tex]\[ y = -2 \][/tex]

Among the given options, the correct choice is:

[tex]\[ y = -2 \][/tex]