Question Video: Finding the Limit of a Piecewise-Defined Function at a Point Mathematics

Video Transcript

Find the limit as 𝑥 approaches the square root of five of 𝑓 of 𝑥 if the function 𝑓 of 𝑥 is equal to seven over two times 𝑥 to the fifth power if 𝑥 is less than the square root of five and 𝑓 of 𝑥 is equal to 𝑥 to the seventh power minus 125 root five divided by 𝑥 squared minus five if 𝑥 is greater than the square root of five.

In this question, we’re given a piecewise-defined function 𝑓 of 𝑥. And we need to use this piecewise definition to determine the limit as 𝑥 approaches root five of 𝑓 of 𝑥. And to do this, we need to notice something interesting about the limit we’re asked to find. We’re asked to find the limit as 𝑥 approaches root five of this function, and we can see this is the endpoint of the subdomains. This means that our function 𝑓 of 𝑥 has a different definition to the left of 𝑥 is equal to root five and to the right of 𝑥 is equal to root five. So we’re going to want to consider the case where 𝑥 approaches root five from the left and the case where 𝑥 approaches root five from the right separately.

We can do this by recalling we say the limit as 𝑥 approaches 𝑎 of some function 𝑓 of 𝑥 is equal to a finite value of 𝐿 if the limit as 𝑥 approaches 𝑎 from the right of 𝑓 of 𝑥 and the limit as 𝑥 approaches 𝑎 from the left of 𝑓 of 𝑥 both exist and are both equal to this finite value of 𝐿. And this property is what allows us to determine the limit of a function by looking at its left and right limits. We need to check that its right limit exists and is equal to 𝐿, and we also need to check that its left limit exists and is also equal to 𝐿. If all of these things hold, then we can say the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝐿. However, if one or more of these don’t hold, then we say the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 does not exist.

We want to apply this to determine the limit as 𝑥 approaches root five 𝑥. So let’s set our value of 𝑎 equal to root five. This then gives us the following. We need to determine the limit as 𝑥 approaches root five from the right of 𝑓 of 𝑥 and the limit as 𝑥 approaches root five from the left of 𝑓 of 𝑥. Let’s start by finding the limit as 𝑥 approaches root five from the right of 𝑓 of 𝑥. Remember, this means our values of 𝑥 are going to be greater than root five since 𝑥 is approaching root five from the right. And we can see when our values of 𝑥 are greater than root five, 𝑓 of 𝑥 is exactly equal to its second subfunction. And if these two functions are equal when 𝑥 is greater than root five, then their limits as 𝑥 approaches root five from the right must be the same. We need to find the limit as 𝑥 approaches root five from the right of 𝑥 to the seventh power minus 125 root five divided by 𝑥 squared minus five.

And this is a rational function. So we can attempt to do this by using direct substitution. Substituting 𝑥 is equal to root five into our rational function, we get root five to the seventh power minus 125 root five divided by root five squared minus five. And we can then evaluate the numerator and denominator of this expression separately. We get zero divided by zero, which is an indeterminate form.

And remember, we can’t conclude that the limit doesn’t exist. Instead, all this tells us is that we need to use a different method to evaluate this limit. And in fact, there are two different methods we can use to evaluate this limit. The first method we can use is to note that we’ve substituted 𝑥 is equal to root five into this rational function. And when we evaluated, we got zero divided by zero. In particular, this tells us root five is a root of the polynomial in the numerator, and it’s also a root of the polynomial in the denominator. So the remainder theorem tells us that 𝑥 minus root five must be a factor of both of these polynomials.

So we can take a factor of 𝑥 minus root five outside of the denominator by using a difference between squares. And we can also do this in the numerator by using one of several methods. For example, we could do this by using algebraic division. This then gives us the following limit. Now since our values of 𝑥 are approaching root five from the right, our values of 𝑥 are never equal to root five. This means that 𝑥 minus root five divided by 𝑥 minus root five is just equal to one.

So now we have the limit of another rational function. We can then evaluate this limit by using direct substitution. This time, we will not get an indeterminate form. And even if we did, we could keep taking out factors of 𝑥 minus root five. And this is one method of evaluating this limit. However, there are a few problems. For example, it’s quite difficult to keep using algebraic division.

So instead, let’s go through a second method which is much easier to calculate. We just need to notice 125 root five is root five to the seventh power and five is root five squared. We can then notice that this limit is in the form of a limit of a difference between powers. Therefore, we can evaluate this limit by using this result. For any real constants 𝑎, 𝑛, and 𝑚, the limit as 𝑥 approaches 𝑎 of 𝑥 to the power of 𝑛 minus 𝑎 to the power of 𝑛 divided by 𝑥 to the power of 𝑚 minus 𝑎 to the power of 𝑚 is 𝑛 divided by 𝑚 multiplied by 𝑎 to the power of 𝑛 minus 𝑚.

And this holds, provided 𝑎 to the power 𝑛, 𝑎 to the power of 𝑚, and 𝑎 to the power of 𝑛 minus 𝑚 all exist and 𝑚 is nonzero. For our limit, the value of 𝑎 is the square root of five, the value of 𝑚 is two, and the value of 𝑛 is seven. So we can evaluate this limit by substituting these values into our limit result. We get seven over two multiplied by root five to the power of seven minus two. We can then evaluate this limit. Seven times root five to the power of five is 175 root five. And then we divide this value by two. Therefore, we’ve shown the limit as 𝑥 approaches the square root of five from the right exists, and we’ve shown that it’s equal to 175 root five over two.

We now need to check the limit as 𝑥 approaches root five from the left of 𝑓 of 𝑥 exists. And we also need to check that it’s equal to the same value of 175 root five over two. So let’s now evaluate the limit as 𝑥 approaches root five from the left of 𝑓 of 𝑥. Remember, our values of 𝑥 are going to be less than the square roots of five. And in this case, our function 𝑓 of 𝑥 is equal to its first subfunction. So once again, the limit of these two functions as 𝑥 approaches root five from the left will be equal. We just need to find the limit as 𝑥 approaches root five from the left of seven over two times 𝑥 to the fifth power.

And this is just a polynomial. So we can just do this by using direct substitution. We substitute 𝑥 is equal to root five into our polynomial. This gives us seven over two times root five to the fifth power. And if we evaluate this expression, we get 175 root five over two, which is exactly the same as our right limit. Therefore, we’ve shown the limit as 𝑥 approaches root five from the left exists, and we’ve also shown that it’s equal to the same value of 𝐿. Therefore, since the left and right limit of 𝑓 of 𝑥 as 𝑥 approaches root five exist and they’re equal, we were able to show the limit as 𝑥 approaches root five of 𝑓 of 𝑥 is equal to 175 over two multiplied by root five.