1.8 Determining Limits Using the Squeeze Theorem

The Squeeze Theorem 🥪

The Squeeze Theorem (or Sandwich Theorem) is used on limit problems where the usual algebraic methods (factoring, conjugation, algebraic manipulation, or trig identities) are not effective. 🥪

The actual Squeeze Theorem states that f(x) ≤ g(x) ≤ h(x) for all x in an open interval about a. If the limit as x approaches a of f(x) = L and the limit as x approaches a of h(x) = L, then the limit as x approaches a of g(x) = L.

With the Squeeze Theorem, we are trying to find two functions that are 1) similar enough to the original function that we can be sure the squeeze works and 2) easier to evaluate their limit as x → a. The theorem utilizes the statement “If two functions squeeze together at a particular point, then any function trapped between them will get squeezed to that same point.” 

So, in order to use the squeeze theorem on a limit, we just have to find functions similar enough that all three functions squeeze together at a particular point like the image below. 

Math Warehouse

If you’re confused, take a look at an example problem using the Squeeze Theorem. 

Example Problem 💡

As you can see in the image above, while the limit of the function in red may be hard to solve for, the similar functions in purple help us figure out that the limit of the function will still be zero. We can use this as an example. 

First, we must confirm that f(x) ≤ g(x) ≤ h(x). From the graph, we can confirm that f(x) and h(x) “squeeze” g(x) at x = 0. Now we know that f(x) ≤ g(x) ≤ h(x), the limit as x approaches 0 of f(x)= 0, and the limit as x approaches 0 of h(x) = 0 using the graph. Thus, the limit as x approaches 0 of g(x) IS also equal to 0.