Question Video: Finding the Limit of a Composition of Rational and Root Functions at Infinity Mathematics

Video Transcript

Find the limit of the square root of 16𝑥 plus eight over nine 𝑥 plus three as 𝑥 approaches infinity.

We just write down the limit we have to evaluate again. The first law of limits that we use is that the limit of the square root of a function is the square root of the limit of the function. In fact, this is true more generally for 𝑛th roots. Applying this law of limits, we see that we just have to find the limit of the rational function 16𝑥 plus eight over nine 𝑥 plus three as 𝑥 approaches infinity. Having found that our limit is just the square root of this, how do we find the limit of our rational function at infinity? At some point, we’ll want to apply the quotient law of limits which says that the limit of a quotient of functions is the quotient of their limits. But if we try to apply this law immediately, something goes wrong. Let’s see what.

Applying this law underneath the radical sign, we get the limit of 16𝑥 plus eight as 𝑥 approaches infinity over the limit of nine 𝑥 plus three as 𝑥 approaches infinity. Now, what is the value of the limit in the numerator? Well, 𝑥 is approaching infinity. So 𝑥 is getting bigger and bigger without bound. And so we would expect 16𝑥 plus eight to also be getting bigger and bigger without bound. The value of this limit is therefore infinity. And exactly the same thing happens in the denominator. 𝑥 is approaching infinity. So 𝑥 is growing without bound. And so nine 𝑥 and hence nine 𝑥 plus three must also be growing without bound.

That slightly handwavy reasoning can be made rigorous. But we’re not going to bother doing that. The point is that underneath the radical sign we have the indeterminate form infinity over infinity. And the indeterminate form infinity over infinity gives us no clue as to what the value of the limit actually is. You can’t just cancel the infinities in the numerator and the denominator to say that the limit is the square root of one. Unfortunately, it’s not quite that simple.

If we get an indeterminate form when trying to find the value of some limit, then the only thing we can do is to go back and try some other method and hope that we don’t get an indeterminate form. Well, actually is not quite that hopeless, we have a method for finding the limits of a rational function of 𝑥 as 𝑥 approaches infinity. We rewrite the rational function that we’re trying to find the limit of by dividing both numerator and denominator by the highest power of 𝑥 that appears in the denominator.

What is the highest power of 𝑥 in the denominator? Our denominator is the polynomial nine 𝑥 plus three. And the highest power of 𝑥 that appears in nine 𝑥 plus three is just 𝑥 to the one or 𝑥. We divide both numerator and denominator by 𝑥 then and we can simplify. We can split up the numerator, making 16𝑥 over 𝑥 plus eight over 𝑥. And 16𝑥 over 𝑥 is just 16. In a similar way, we can simplify the denominator, getting nine plus three over 𝑥.

Now, we can apply the quotient law of limits. And I claim that this time we won’t get an indeterminate form. We get the square root of the limit of 16 plus eight over 𝑥 as 𝑥 approaches infinity over the limit of nine plus three over 𝑥 as 𝑥 approaches infinity. Now, what’s the value of the limit in the numerator? Well, as 𝑥 is getting bigger and bigger, eight over 𝑥 is getting smaller and smaller, getting closer and closer to zero. And so 16 plus eight over 𝑥 is getting closer and closer to 16.

We could show this rigorously by using the fact that the limit of a sum of functions is the sum of their limits, that the limit of a constant function like 16 is just the constant itself, and the limit of a number times a function is that number times the limit of the function. This allows us to write this limit in terms of the limit of the reciprocal function one over 𝑥 as 𝑥 approaches infinity. And it’s another law of limits that the limit of the reciprocal function as 𝑥 approaches plus or minus infinity is just zero. And so our limit is 16 plus eight times zero which is 16.

Having found the limit in the numerator to be 16, we can do something very similar in the denominator. As 𝑥 gets bigger and bigger without bound, three over 𝑥 gets smaller and smaller, closer and closer to zero. And so nine plus three over 𝑥 gets closer and closer to nine. The value of our limit is, therefore, the square root of 16 over nine. And as both 16 and nine are squares, 16 is four squared and nine is three squared. We can simplify to get our final answer four-thirds.

The key to answering this question is the key to answering all questions about the limits of rational functions as 𝑥 approaches either infinity or minus infinity. That is, before applying the quotient law of limits, you should first divide both numerator and denominator of your rational function by the highest power of 𝑥 that appears in the polynomial that forms the denominator of your rational function.