Squeeze Theorem – Sandwich Theorem

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:

squeeze theorem

For both functions, when x = 0, y = 0.

Let’s “squeeze” y = sin(x^2) in between both functions.

sandwich theorem

theorem 1

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.

squeeze

sandwich

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:

functions

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.

squeeze theorem 1

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:

functions 1-1