Squeeze theorem
If a function is trapped between two functions that have the same limit, it must have that limit too.
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Analogy
A sandwich: if both slices end up at the same spot, the filling cannot go anywhere else.
Example
Since -1 <= sin(1/x) <= 1:-x^2 <= x^2 sin(1/x) <= x^2.Both -x^2 and x^2 tend to 0.So the limit is 0.
In practice
It handles oscillating terms like sin(1/x), which have no limit by themselves.
Practice
Flashcards
- State the squeeze theorem in one sentence.If g <= f <= h and g and h share limit L, then f has limit L.
Questions people ask
When should I use the squeeze theorem?
Use it when a function has a bounded wobbling part, such as sine or cosine, multiplied by something tending to zero. You bound the wobble between -1 and 1, multiply through, and show both outer functions share a limit. Then your middle function must share it too.
Why does the squeeze theorem work?
If g is below f and h is above f, and both g and h close in on the same number L, then f has nowhere else to go. It is stuck between two walls that meet at L. This is why it is also called the sandwich theorem.