Question Video: Using the Squeeze Theorem on Polynomials at a Point Mathematics

Video Transcript

Using the squeeze theorem, check whether the following statement is true or false. If some function 𝑔 of 𝑥 is greater than or equal to three 𝑥 minus three and 𝑔 of 𝑥 is less than or equal to two 𝑥 squared minus four 𝑥 plus three, then the limit as 𝑥 approaches two of 𝑔 of 𝑥 is equal to zero.

The question gives us a statement about some function 𝑔 of 𝑥. And it wants us to use the squeeze theorem to determine whether this means the limit as 𝑥 approaches two of 𝑔 of 𝑥 will be equal to zero. So let’s start by recalling what the squeeze theorem tells us.

The squeeze theorem tells us if 𝑔 of 𝑥 is greater than or equal to some function 𝑓 of 𝑥 and 𝑔 of 𝑥 is less than or equal to some function ℎ of 𝑥. For all values of 𝑥 near a constant 𝑎 but not necessarily at 𝑥 is equal to 𝑎. And if we also know the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 and the limit as 𝑥 approaches 𝑎 of ℎ of 𝑥 are both equal to some finite constant 𝐿. Then the squeeze theorem tells us the limit as 𝑥 approaches 𝑎 of our function 𝑔 of 𝑥 must also be equal to 𝐿.

So we need to work out how we’re going to apply the squeeze theorem to this question. We can see our function 𝑔 of 𝑥 has an upper bound and a lower bound. And we can see this is the same as the function 𝑔 of 𝑥 in the squeeze theorem. So our concluding statement in the squeeze theorem tells us about the limit as 𝑥 approaches 𝑎 of 𝑔 of 𝑥.

We want to make a conclusion about the limit as 𝑥 approaches two of 𝑔 of 𝑥. So we should take our value of 𝑎 equal to two. So let’s rewrite our squeeze theorem with 𝑎 equal to two. Next, we can see that our function 𝑓 of 𝑥 is the linear function three 𝑥 minus three, and our function ℎ of 𝑥 is the quadratic two 𝑥 squared minus four 𝑥 plus three.

To use the squeeze theorem, we need to calculate the limit as 𝑥 approaches two of 𝑓 of 𝑥 and the limit as 𝑥 approaches two of ℎ of 𝑥. Let’s start with the limit as 𝑥 approaches two of our linear function three 𝑥 minus three. We can see this is a linear function. So we can just do this by using direct substitution. Substituting 𝑥 is equal to two, we get three times two minus three. And we can calculate this. It’s equal to three.

Let’s now do the same with our upper bound for 𝑔 of 𝑥, the function two 𝑥 squared minus four 𝑥 plus three. Again, we can calculate this limit by using direct substitution. Substituting 𝑥 is equal to two, we get two times two squared minus four multiplied by two plus three. And we can evaluate this. We see it’s equal to three.

So when we evaluated both of these limits, we got the same value of three. So our value of 𝐿 in the squeeze theorem is equal to three. So let’s think about what we’ve shown. The question tells us that we have upper and lower bounds for our function 𝑔 of 𝑥. And we’ve shown that the limit as 𝑥 approaches two for both of these functions is equal to three. So by the squeeze theorem, we can conclude the limit as 𝑥 approaches two of our function 𝑔 of 𝑥 must be equal to three. Of course, this means the limit as 𝑥 approaches two of our function 𝑔 of 𝑥 cannot be equal to zero.

Therefore, by using the squeeze theorem, we were able to show that if 𝑔 of 𝑥 is greater than or equal to three 𝑥 minus three and 𝑔 of 𝑥 is less than or equal to two 𝑥 squared minus four 𝑥 plus three. Then the limit as 𝑥 approaches two of 𝑔 of 𝑥 cannot be equal to zero.