Answer:
[tex]\lim _{x\to-\infty}f(x)=\lim _{x\to\infty}f(x)=-2[/tex]Explanation:
Given a rational function, the end behavior of its graph is how the graph behaves as x approaches infinity or negative infinity.
In the function f(x) below:
[tex]f(x)=-\frac{2x}{x-6}[/tex]• The degree of the numerator = 1
,• The degree of the denominator = 1
Since the degrees of the numerator and the denominator are the same, divide the coefficients of the leading terms to obtain the horizontal asymptote.
[tex]\text{Horizontal Asymptote, }y=-\frac{2}{1}=-2[/tex]Thus, for f(x):
[tex]\begin{gathered} \lim _{x\to-\infty}f(x)=-2 \\ \lim _{x\to\infty}f(x)=-2 \end{gathered}[/tex]The function approaches -2 as x tends to positive and negative infinity.
Next, we determine the vertical asymptote of f(x) by setting the denominator equal to 0 and solving for x.
[tex]\begin{gathered} x-6=0 \\ x=6 \end{gathered}[/tex]What this means is that in f(x), the value of f(x) is undefined at x=6.
