Limits & Continuity · Continuity & Theorems
Squeeze Theorem
If f(x) is squeezed between g(x) and h(x) near a, and g and h have the same limit L, then f also has limit L.
- 2 worked examples
- 18 in Limits & Continuity
Worked examples
- We know -1 ≤ sin(1/x) ≤ 1 for all x ≠ 0
- Multiply by x²: -x² ≤ x² sin(1/x) ≤ x²
- lim(x→0) -x² = 0 and lim(x→0) x² = 0
- By Squeeze Theorem, lim(x→0) x² sin(1/x) = 0
Answer: 0
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Common questions
What is the Squeeze Theorem?
If f(x) is squeezed between g(x) and h(x) near a, and g and h have the same limit L, then f also has limit L. It is one of the limits & continuity formulas in the CalcRef reference.
How do you use the Squeeze Theorem?
Worked example. Find lim(x→0) x² sin(1/x). We know -1 ≤ sin(1/x) ≤ 1 for all x ≠ 0. Multiply by x²: -x² ≤ x² sin(1/x) ≤ x². lim(x→0) -x² = 0 and lim(x→0) x² = 0. By Squeeze Theorem, lim(x→0) x² sin(1/x) = 0. Answer: 0.
Practice
Quiz yourself on Limits & Continuity
Squeeze Theorem is one of 18 Limits & Continuity formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.