# Squeeze Theoremap Calculus

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:

The squeeze (or sandwich) theorem states that if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them. Copter game. We can use the theorem to find tricky limits like sin(x)/x at x=0, by 'squeezing' sin(x)/x between two nicer functions and using them to find the limit at x=0.

• Assignments AB 2020-2021. 1.01 Numerical Limits. 1.02 Graphical Limits. 1.03 Algebraic Limits. 1.04 Squeeze Theorem.
• The Squeeze Principle is used on limit problems where the usual algebraic methods (factoring, conjugation, algebraic manipulation, etc.) are not effective. However, it requires that you be able to ``squeeze' your problem in between two other ``simpler' functions whose limits are easily computable and equal.

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 sin(x), it doesn’t matter what x is, is between -1 and 1. We multiply the inside, f(x), by x^2, to get our original function. We multiply the outside functions, g(x) and h(x), by x^2 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:
g(x)=-x^2
f(x)=x^2sin(frac{1}{x^2})
h(x)=x^2