The Squeeze Theorem, also known as the Sandwich Theorem, is a powerful tool in calculus for determining the limit of a function. It states that if two functions “squeeze” a third function between them, and the two outer functions have the same limit at a point, then the middle function must also have the same limit at that point.
Formal Statement
Let $ f(x) $ , $ g(x) $ , and $ h(x) $ be functions defined on an open interval containing $ a $ , except possibly at $ a $ itself. If:
- $ g(x) \leq f(x) \leq h(x) $ for all $ x $ in the interval (except possibly at $ x=a $ )
- $ \lim_{x \to a} g(x) = L $ and $ \lim_{x \to a} h(x) = L $
Then:
$ \lim_{x \to a} f(x) = L $
Desmos example
Example
Let’s find the limit of the function $ f(x) = x^2 \sin(\frac{1}{x}) $ as $ x $ approaches 0.
1. Find bounding functions:
We know that $ -1 \leq \sin(\frac{1}{x}) \leq 1 $ for all $ x $ (except $ x=0 $ ). Multiplying this inequality by $ x^2 $ , we get:
$ -x^2 \leq x^2 \sin(\frac{1}{x}) \leq x^2 $
2. Find the limits of the bounding functions:
$ \lim_{x \to 0} -x^2 = 0 $ and $ \lim_{x \to 0} x^2 = 0 $
3. Apply the Squeeze Theorem:
Since $ -x^2 \leq x^2 \sin(\frac{1}{x}) \leq x^2 $ and both $ -x^2 $ and $ x^2 $ approach 0 as $ x $ approaches 0, we can conclude that:
$ \lim_{x \to 0} x^2 \sin(\frac{1}{x}) = 0 $
Applications
The Squeeze Theorem is particularly useful when dealing with functions that are difficult to evaluate directly. It is often used to:
- Find limits involving trigonometric functions.
- Prove the limit of a sequence.
- Determine the convergence of a series.
Key Points
- The Squeeze Theorem only works if the bounding functions have the same limit.
- The inequality $ g(x) \leq f(x) \leq h(x) $ must hold for all values of $ x $ in the interval, except possibly at $ x=a $ .
- The Squeeze Theorem can be used to find limits of functions that are otherwise difficult to evaluate.
By understanding the Squeeze Theorem, you gain a valuable tool for solving a variety of calculus problems.