Squeeze Theorem: Definition, Visual Example & How It Works
The green curve keeps wiggling near 0, so chasing its exact value point by point is hopeless. Instead, trap it: yellow x² stays above, pink −x² stays below. Drag right and watch both walls collapse toward 0.
Move the idea
Try it: shrink the trap around x=0
Change one parameter and watch what changes with it.
Stop chasing the wiggle. Control its cage. If both walls squeeze to the same number, everything trapped between them is forced to go there too.
Do not chase the wiggle
Because −1≤sin(10/x)≤1, multiplying by x² gives −x²≤x²sin(10/x)≤x². The green curve can oscillate as much as it wants, but it can never escape the two parabolas.
Watch the cage shrink
Inside |x|≤δ, both walls are at most δ² away from zero. Make δ smaller and the vertical room left for the green curve collapses toward zero.
The reusable rule
The Squeeze Theorem is a comparison principle: when a hard function stays between two easier functions with the same limit, the middle limit is forced too.
Make the idea reusable.
Squeeze Theorem statement
If g(x)≤f(x)≤h(x) near x=a and both outer functions approach the same limit L, then f(x) must also approach L.
When to use it
Look for a difficult function whose size or oscillation can be bounded by simpler expressions. Trigonometric factors such as sin(1/x) are classic candidates because their values stay between −1 and 1.
What the theorem does not require
The middle function does not need to be monotone, and its value at x=a does not determine the limit. The key requirement is that both valid bounds converge to the same destination.