What happens when algebraic manipulation does not work to find the limit? Give the squeeze theorem, also known as the sandwich theorem, a try! The squeeze theorem helps you find the limit of a function by comparing the limits of two simpler functions that are the lower and upper bounds.
The Squeeze Theorem:
What does the Squeeze Theorem mean?
Given a function, f(x), take two simpler functions, g(x) and h(x), that are a higher and lower bound of f(x). If the limit of g(x) and h(x) as x approaches c are the same, then the limit of f(x) as x approaches c must be the same as their limit because f(x) is squeezed, or sandwiched, between them.
Here is an image to help better understand the theorem:
Here we will work out the first problem step by step (click here):
1. Try Substitution
When we substitute 0 for x, we get an undefined answer.
2. Find g(x)and h(x)
We know that , it doesn’t matter what x is, is between -1 and 1. We multiply the inside, f(x), by , to get our original function. We multiply the outside functions, g(x) and h(x), by too.
3. Substitution for the outer limits
We substitute in 0 for x in g(x) and h(x) to find their limits. Since their limits as x approaches 0 both equal 0, then by the squeeze theorem, the limit of f(x) as x approaches 0 is also 0.
Here is an image to better understand the solution to the problem:
Here’s another example (click here):
Last example (click here):
The squeeze theorem is a very useful theorem to quickly find the limit. However, finding the upper and lower bound functions can be hard. Sometimes graphing f(x) in order to see what the function approaches at x can be helpful when deciding what the lower and upper bounded functions should be.
Until Next Time,