Find the limit as 𝑥 approaches nine from the right of 𝑥 squared plus 18𝑥 plus 81 divided by 𝑥 squared minus seven 𝑥 minus 18.
In this question, we’re asked to evaluate the limit of a rational function, in this case it’s the quotient of two quadratics. And we can recall we can always try and evaluate the limit of rational functions by using direct substitution. This means we can substitute 𝑥 is equal to nine into our rational function. And if this evaluates to give us a real number, then we can conclude that this is the limit.
So by substituting nine into this rational function, we get nine squared plus 18 times nine plus 81 all divided by nine squared minus seven times nine minus 18. And if we evaluate this expression, the numerator gives us 324 and the denominator gives us zero. We might then be tempted to conclude that this is an indeterminate form. However, this is not the case. A nonzero integer divided by zero is not an indeterminate form; zero divided by zero is an indeterminate form.
Therefore, we can actually determine information about our limit from this substitution. Since this is the limit of a rational function and the numerator of this expression is approaching a constant value as 𝑥 approaches nine and the denominator of this rational function approaches zero, we can conclude the outputs are unbounded as 𝑥 approaches nine. Therefore, our rational function will have a vertical asymptote at 𝑥 is equal to nine.
Of course, this doesn’t directly let us evaluate this limit because there are several different possible types of vertical asymptote. The function could approach positive ∞ as 𝑥 approaches nine from both sides, negative ∞ as 𝑥 approaches nine from both sides, or it could approach positive ∞ and negative ∞ on either side. This gives us four possible options for what’s happening to our function around 𝑥 is equal to nine. However, we’re only interested in what happens to the outputs of our function as 𝑥 approaches nine from the right, and we can determine this in many different ways.
The easiest way will be to consider the limit of the numerator and denominator separately. We can then combine these at the end. First, we’ve already shown the limit of the numerator as 𝑥 approaches nine from the right is 324. That’s a positive number. Next, we want to evaluate the limit as 𝑥 approaches nine from the right of the denominator. That’s 𝑥 squared minus seven 𝑥 minus 18. Of course, we can substitute 𝑥 is equal to nine into this quadratic to see that this is zero. However, this is not going to allow us to evaluate this limit. Instead, we need to know what happens when 𝑥 is close to nine from the right.
To do this, let’s sketch the graph of 𝑥 squared minus seven 𝑥 minus 18. To help us sketch this graph, we start by factoring the quadratic. It’s equal to 𝑥 minus nine multiplied by 𝑥 plus two. So this is a quadratic with roots at negative two and nine with positive leading coefficient. So it opens upwards. We can then use this to determine what happens to our outputs as 𝑥 approaches nine from the right.
The outputs of this function are approaching zero from the positive side; the outputs are always positive. Therefore, our outputs are approaching zero from the right. And this is really useful when we’re trying to evaluate the limit of our rational function. As 𝑥 approaches nine from the right, our numerator approaches the positive value of 324. And our denominator is approaching zero from the right; it’s also always a positive number. Therefore, the outputs of this function as 𝑥 approaches nine from the right is always a positive number. It’s the quotient of two positive numbers.
And either by using the fact we have a vertical asymptote at this point or the fact that our numerator remains constant, however, our denominator is approaching zero — so we know the limit must be unbounded — we can conclude the outputs of this function are increasing without bound. Its limit must be ∞. Therefore, we were able to show the limit as 𝑥 approaches nine from the right of 𝑥 squared plus 18𝑥 plus 81 divided by 𝑥 squared minus seven 𝑥 minus 18 is ∞.