The Squeeze Theorem can be summarized into two main points:
- If two functions “squeeze” together at a particular point, then any function trapped between them will get “squeezed” to that same point.
- The Squeeze Theorem is used to find limits of functions, not numerical values of functions.
Graphical Example
Let’s say that we wanted to find the limit of y = sin(x^2) when x approaches 0. Obviously, you can plug in x = 0 into the equation and get y = sin (0) = 0. However, let’s say that you wanted to use the Squeeze Theorem to prove this.
Let’s use two other functions, y = x^2 and y = -x^2. Let’s graph this on the coordinate plane:
For both functions, when x = 0, y = 0.
Let’s “squeeze” y = sin(x^2) in between both functions.
Notice that at x=0, the limit of y = sin(x^2) is also zero. Therefore, the Squeeze Theorem is as follows:
If f(x)≤g(x)≤h(x) for all x in an open interval about a (except possibly at a itself)
and
limit of f(x) x→a = limit of h(x)x→a = L,
then limit g(x) x→a = L<>
In other words, f(x) and h(x) are the top and bottom functions, respectively, squeezing the middle function, g(x). All three functions meet at the same point. Therefore, all three functions have a limit at that same point.
Example 1 – Find the limit of x^2*sin(1/x) when x approaches zero.
Step 1: Find x^2 sin(1/x) at x = 0.
(0)^2 * sin(1/0) = undefined.
When we look at the function at x=0, we can see why it is undefined. The graphs oscillate back and forth with increased frequency as x approaches zero, but we are not sure if it reaches zero at x=0.
Step 2: Pick f(x)
Let’s choose f(x) = x^2
f(0) = 0^2 = 0.
Step 3: Pick h(x)
Let’s choose h(x) = -x^2
f(0) = -(0)^2 = 0.
Let’s graph all three functions on the coordinate plane:
According to the Squeeze Theorem, the limit of x^2 * sin(1/x) as x approaches zero is zero.
Example 2 – If f(x) = 4-x^2 and h(x) = 4+ x^2, find the limit of g(x) when x approaches zero.
Step 1: find f(0)
f(0) = 4-(0)^2 = 4-0 = 4.
Step 2: find h(0)
h(0) = 4+(0)^2 = 4+0 = 4.
Step 3: graph f(x) and h(x) on the coordinate plane to see if they “squeeze” against each other.
Since the squeeze against each other at the point (0,4), any function, g(x), that is in between these functions, will also have a limit at that point. An example of g(x) is y = 4:
