Question

Show all work to identify the asymptotes and state the end behavior of the function f(x) = 6x/ x - 36

Answer 1

Vertical asymptotes are when the denominator is 0 but the numerator isn't 0.

$x-36=0 \implies x=36$

Since this value of x does not make the numerator equal to 0, the vertical asymptote is $x=36$.

Horizontal asymptotes are the limits as $x \to \pm \infty$.

$\lim_{x \to \infty} \frac{6x}{x-36}=\lim_{x \to \infty}=\frac{6}{1-\frac{36}{x}}=6\\\\\lim_{x \to -\infty} \frac{6x}{x-36}=\lim_{x \to -\infty}=\frac{6}{1-\frac{36}{x}}=6$

So, the horizontal asymptote is $y=6$.

End behavior:

• As $x \to \infty, f(x) \to -\infty$
• As $x \to -\infty, f(x) \to \infty$.