# How do you find the Vertical, Horizontal, and Oblique Asymptote given g(t) = (t − 6) / (t^(2) + 36)?

Aug 7, 2018

#### Answer:

This function only has a horizontal asymptote $y = 0$

#### Explanation:

Given:

$g \left(t\right) = \frac{t - 6}{{t}^{2} + 36}$

Note that the denominator is positive for any real value of $t$.

Hence this rational function has no vertical asymptotes and no holes.

The degree of the denominator is also greater than the numerator. So it has no blique asymptotes - only the horizontal asymptote $y = 0$.

${\lim}_{t \to \pm \infty} \frac{t - 6}{{t}^{2} + 36} = {\lim}_{t \to \pm \infty} \frac{\frac{1}{t} - \frac{6}{t} ^ 2}{1 + \frac{36}{t} ^ 2} = \frac{0 - 0}{1 + 0} = 0$

graph{(x-6)/(x^2+36) [-20, 30, -0.3, 0.3]}