# How do I identify the horizontal asymptote of f(x) = (7x+1)/(2x-9)?

Mar 9, 2018

We have a horizontal asymptote $y = 3.5$

#### Explanation:

As the degree of polynomial in the numerator is equal to the degree of polynomial in the denominator, there is indeed a horizontal asymptote. We can find this by dividing each term in numerator and denominator by this highest degree and find limit as $x \to \infty$. The process is shown below:

Now ${\lim}_{x \to \infty} \frac{7 x + 1}{2 x - 9}$

= ${\lim}_{x \to \infty} \frac{7 + \frac{1}{x}}{2 - \frac{9}{x}}$

= $\frac{7}{2}$

Hence, we have a horizontal asymptote $y = \frac{7}{2}$ or $y = 3.5$

graph{(y-(7x+1)/(2x-9))(y-3.5)=0 [-40.42, 39.58, -17.76, 22.24]}