Question

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

Answer 1

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->oo`. The process is shown below:

Now `lim_(x->oo)(7x+1)/(2x-9)`

= `lim_(x->oo)(7+1/x)/(2-9/x)`

= `7/2`

Hence, we have a horizontal asymptote `y=7/2` or `y=3.5`

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