Question

question
what is the horizontal asymptote of the graph of ( f(x)=\frac{-2 x}{x - 3} )?
give your answer in the form ( y = a ).
provide your answer below:
( y = )

Explanation:

Step1: Divide numerator and denominator by x

For \( f(x)=\frac{-2x}{x - 3} \), divide numerator and denominator by \( x\) (where \(x
eq0\)). We get \( f(x)=\frac{-2}{\frac{x}{x}-\frac{3}{x}}=\frac{-2}{1-\frac{3}{x}}\).

Step2: Find the limit as \(x\to\pm\infty\)

As \(x\to\pm\infty\), \(\lim_{x\to\pm\infty}\frac{3}{x}=0\). Then \(\lim_{x\to\pm\infty}f(x)=\lim_{x\to\pm\infty}\frac{-2}{1 - \frac{3}{x}}\). Substituting \(\lim_{x\to\pm\infty}\frac{3}{x}=0\) into the expression, we have \(\frac{-2}{1-0}=-2\).

Answer:

\(y = - 2\)