# Question Video: Using the Squeeze Theorem on Polynomials at a Point Mathematics • Higher Education

Using the squeeze theorem, check whether the following statement is true or false: If 3π₯ β 3 β€ π(π₯) β€ 2π₯Β² β 4π₯ + 3, then lim_(π₯ β 2) π(π₯) = 0.

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### Video Transcript

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

The question gives us a statement about some function π of π₯. And it wants us to use the squeeze theorem to determine whether this means the limit as π₯ approaches two of π of π₯ will be equal to zero. So letβs start by recalling what the squeeze theorem tells us.

The squeeze theorem tells us if π of π₯ is greater than or equal to some function π of π₯ and π of π₯ is less than or equal to some function β of π₯. For all values of π₯ near a constant π but not necessarily at π₯ is equal to π. And if we also know the limit as π₯ approaches π of π of π₯ and the limit as π₯ approaches π of β of π₯ are both equal to some finite constant πΏ. Then the squeeze theorem tells us the limit as π₯ approaches π of our function π of π₯ must also be equal to πΏ.

So we need to work out how weβre going to apply the squeeze theorem to this question. We can see our function π of π₯ has an upper bound and a lower bound. And we can see this is the same as the function π of π₯ in the squeeze theorem. So our concluding statement in the squeeze theorem tells us about the limit as π₯ approaches π of π of π₯.

We want to make a conclusion about the limit as π₯ approaches two of π of π₯. So we should take our value of π equal to two. So letβs rewrite our squeeze theorem with π equal to two. Next, we can see that our function π of π₯ is the linear function three π₯ minus three, and our function β of π₯ is the quadratic two π₯ squared minus four π₯ plus three.

To use the squeeze theorem, we need to calculate the limit as π₯ approaches two of π of π₯ and the limit as π₯ approaches two of β of π₯. Letβs start with the limit as π₯ approaches two of our linear function three π₯ minus three. We can see this is a linear function. So we can just do this by using direct substitution. Substituting π₯ is equal to two, we get three times two minus three. And we can calculate this. Itβs equal to three.

Letβs now do the same with our upper bound for π of π₯, the function two π₯ squared minus four π₯ plus three. Again, we can calculate this limit by using direct substitution. Substituting π₯ is equal to two, we get two times two squared minus four multiplied by two plus three. And we can evaluate this. We see itβs equal to three.

So when we evaluated both of these limits, we got the same value of three. So our value of πΏ in the squeeze theorem is equal to three. So letβs think about what weβve shown. The question tells us that we have upper and lower bounds for our function π of π₯. And weβve shown that the limit as π₯ approaches two for both of these functions is equal to three. So by the squeeze theorem, we can conclude the limit as π₯ approaches two of our function π of π₯ must be equal to three. Of course, this means the limit as π₯ approaches two of our function π of π₯ cannot be equal to zero.

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