How do you find the horizontal asymptote for fR(x)=(3x+5) /(x-6)?

Mar 16, 2018

The horizontal asymptote is $y = 3$

Explanation:

Calculate the limit as the function $f \left(x\right)$ tend to $+ \infty$ and $- \infty$

The function is

$f \left(x\right) = \frac{3 x + 5}{x - 6}$

The domain of $f \left(x\right)$ is $\mathbb{R} - \left\{6\right\}$

$f \left(x\right) = \frac{3 x + 5}{x - 6} = \frac{\cancel{x} \left(3 + \frac{5}{x}\right)}{\cancel{x} \left(1 - \frac{6}{x}\right)} = \frac{3 + \frac{5}{x}}{1 - \frac{6}{x}}$

The limits are

${\lim}_{x \to + \infty} \left(\frac{5}{x}\right) = 0$

${\lim}_{x \to + \infty} \left(\frac{6}{x}\right) = 0$

${\lim}_{x \to - \infty} \left(\frac{5}{x}\right) = 0$

${\lim}_{x \to - \infty} \left(\frac{6}{x}\right) = 0$

And finally

${\lim}_{x \to + \infty} f \left(x\right) = {\lim}_{x \to + \infty} \frac{3 + \frac{5}{x}}{1 - \frac{6}{x}} = \frac{3 - 0}{1 - 0} = 3$

The horizontal asymptote is $y = 3$

graph{(y-(3x+5)/(x-6))(y-3)=0 [-34.77, 47.43, -16.94, 24.16]}