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?CalculusCalculus
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.
Theoremap2. 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 Squeeze Theoremap Calculusx^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
Here’s another example (click here):

Squeeze Theorem Ap Calculus Worksheet


Last example (click here):

Squeeze Theoremap Calculus Definition

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,

Squeeze Theoremap Calculus Test


Leah

Ap Calculus Squeeze Theorem Worksheet