Squeeze Theorem
If g ≤ f ≤ h and g, h → L then f → L.
Key idea
If g ≤ f ≤ h and g, h → L then f → L.
In depth
If g(x) ≤ f(x) ≤ h(x) near a, and g and h both tend to L, then f tends to L. It traps a hard function between two easy ones.
Worked example
x² sin(1/x) is between −x² and x², both tending to 0, so the limit at 0 is 0.
Formula
Real-world uses
- Proving that sin x / x tends to 1
- Bounding errors in numerical methods
Common mistake
Using bounds that do not converge to the same value.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Squeeze Theorem?
If g ≤ f ≤ h and g, h → L then f → L. It is a topic at the advanced level in Calculus.
What is the formula for Squeeze Theorem?
The key formula is lim sin x / x = 1.
Where is Squeeze Theorem used in real life?
Real-world examples: Proving that sin x / x tends to 1; Bounding errors in numerical methods.
What should I learn before Squeeze Theorem?
Make sure you are comfortable with Infinite Limits and Asymptotes, the previous topic in Calculus.
What is a common mistake with Squeeze Theorem?
Using bounds that do not converge to the same value.
How can I practise Squeeze Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Squeeze Theorem?
Move on to Continuity next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.