Question Video: Using the Squeeze Theorem to Evaluate a Limit Mathematics • Higher Education

Video Transcript

Given that 𝑓 of 𝑥 is greater than or equal to negative three 𝑥 squared plus nine 𝑥 minus five and less than or equal to 𝑥 squared minus three 𝑥 plus four, find the limit as 𝑥 approaches three over two of 𝑓 of 𝑥.

In this question, we are given a function 𝑓 of 𝑥 and told that it is bounded from above by the function 𝑥 squared minus three 𝑥 plus four and bounded from below by the function negative three 𝑥 squared plus nine 𝑥 minus five. We are asked to find the limit as 𝑥 approaches three over two of 𝑓 of 𝑥.

The fact that we are asked to find the limit of a function, which is bounded between two other functions, reminds us of the squeeze theorem. The squeeze theorem says that if there exist functions 𝑔 of 𝑥, 𝑓 of 𝑥, and ℎ of 𝑥, such that 𝑓 of 𝑥 is greater than or equal to 𝑔 of 𝑥 but less than or equal to ℎ of 𝑥. When 𝑥 is near 𝑎, except possibly at 𝑎, and the limit as 𝑥 approaches 𝑎 of 𝑔 of 𝑥 equals the limit as 𝑥 approaches 𝑎 of ℎ of 𝑥 equals 𝐿. Then the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 equals 𝐿.

In order to use the squeeze theorem, we need to have a function 𝑓 of 𝑥 that is bounded from above and below by functions ℎ of 𝑥 and 𝑔 of 𝑥, respectively. We are told that the function 𝑓 of 𝑥 whose limit we are asked to find as 𝑥 approaches three over two is bounded from above by the function 𝑥 squared minus three 𝑥 plus four. And bounded from below by the function negative three 𝑥 squared plus nine 𝑥 minus five. Since the question does not state any particular 𝑥-values for which these bounds hold, we can assume that these bounds hold for all real numbers 𝑥 or at least for values of 𝑥 near three over two.

So letting 𝑔 of 𝑥 equal negative three 𝑥 squared plus nine 𝑥 minus five, ℎ of 𝑥 equal 𝑥 squared minus three 𝑥 plus four, and 𝑎 equal three over two. In the squeeze theorem, we deduce that if the limit as 𝑥 approaches three over two of negative three 𝑥 squared plus nine 𝑥 minus five equals the limit as 𝑥 approaches three over two of 𝑥 squared minus three 𝑥 plus four equals 𝐿. Then the limit as 𝑥 approaches three over two of 𝑓 of 𝑥 equals 𝐿. Let’s evaluate the limit as 𝑥 approaches three over two of negative three 𝑥 squared plus nine 𝑥 minus five and the limit as 𝑥 approaches three over two of 𝑥 squared minus three 𝑥 plus four.

Since negative three 𝑥 squared plus nine 𝑥 minus five and 𝑥 squared minus three 𝑥 plus four are polynomial expressions and the value three over two is a real number. We can evaluate the limits using direct substitution. Doing so, we obtain that the limit as 𝑥 approaches three over two of negative three 𝑥 squared plus nine 𝑥 minus five equals negative three timesed by three over two squared plus nine times three over two minus five, which equals seven over four. We obtain that the limit as 𝑥 approaches three over two of 𝑥 squared minus three 𝑥 plus four equals three over two squared minus three times three over two plus four, which also equals seven over four.

So, we have that the limit as 𝑥 approaches three over two of negative three 𝑥 squared plus nine 𝑥 minus five equals the limit as 𝑥 approaches three over two of 𝑥 squared minus three 𝑥 plus four. And they are both equal to seven over four. So by the squeeze theorem, the limit as 𝑥 approaches three over two of 𝑓 of 𝑥 also equals seven over four. Hence, we have shown, using the squeeze theorem, that if the function 𝑓 of 𝑥 is bounded from above by the function 𝑥 squared minus three 𝑥 plus four and bounded from below by the function negative three 𝑥 squared plus nine 𝑥 minus five. Then the limit as 𝑥 approaches three over two of 𝑓 of 𝑥 equals seven over four.