## Wednesday, July 29, 2015

### Advanced Math Solutions – Limits Calculator, Squeeze Theorem

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 \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^2\sin(\frac{1}{x^2})
h(x)=x^2