AP® Calculus BC · Unit 1 · Topic 1.8
Determining Limits Using the Squeeze Theorem
The Squeeze Theorem says if a function is trapped between two others that both approach the same limit at a point, the trapped function must approach that limit too.
Some functions are too wild to plug values into directly, usually because they oscillate violently near the point you care about. The fix is to sandwich the wild function between two tame ones you already understand. If g(x) ≤ f(x) ≤ h(x) near x = c, and both g and h squeeze down to the same value L as x approaches c, then f has no room to go anywhere else. It gets pinched into L too.
The mechanism is just a trap closing. As x approaches c, the gap between g(x) and h(x) shrinks to zero because both converge to L. Since f(x) is always caught between them, the gap between f(x) and L shrinks to zero as well, even though you never computed f directly. The bounding functions do all the work; f just gets carried along for the ride.
This matters most for limits involving oscillation, like anything with sin(1/x) or cos(1/x) tangled into it, since those terms refuse to settle down near zero no matter how close you get. Multiplying by something that shrinks to zero, like x2, tames the oscillation even though the oscillating part itself has no limit. This topic also sets up later limit techniques: once continuity and limit laws feel automatic, squeezing is the tool you reach for whenever direct substitution and factoring both fail.
Key terms
- Squeeze Theorem
- If g(x) ≤ f(x) ≤ h(x) near c and g, h both approach L as x approaches c, then f also approaches L.
- Bounding functions
- The two tame functions, like g and h, used to trap the wild function from below and above.
- Oscillating function
- A function like sin(1/x) that swings repeatedly between values and never settles as x approaches a point.
- Bounded function
- A function whose values stay between two fixed constants, which alone does not guarantee a limit exists.
Watch it explained
- How it worksHow do two outer curves force the middle curve to land on the very same value?
- Worked exampleFind the limit as x approaches 0 of x2 sin(1/x) by squeezing it between -x2 and x2.
- Common mistakeWhy isn't sin(1/x) having no limit at zero enough to kill x2 sin(1/x) as well?
- Going furtherWhat happens to the squeeze argument when the two bounding curves land on different values?
Check yourself
End of topic.