What the Squeeze Theorem Is For
Sometimes a limit cannot be found by direct substitution, algebraic manipulation, or a familiar limit rule. The function may oscillate, contain a difficult trigonometric expression, or behave in a way that makes its exact value hard to determine directly.
The Squeeze Theorem provides another strategy. Instead of finding the limit of the complicated function directly, you find two simpler functions that trap it from above and below. If both simpler functions approach the same value, the function caught between them must approach that value too.
The basic idea is easiest to visualize as a function being squeezed between two functions:
If the lower and upper functions both approach the same number , then the function in the middle is forced to approach that number as well.
The Squeeze Theorem
Suppose that for all sufficiently close to , except possibly at , we have:
If both outside functions have the same limit, then the middle function has that same limit:
In words: if a function is trapped between two functions that both approach the same value, then the trapped function must approach that value too.
How to Recognize a Squeeze Theorem Problem
The Squeeze Theorem is especially useful when you see a function that is difficult to analyze directly but can be bounded using a familiar inequality.
Look for expressions involving:
- Bounded trigonometric functions, such as and .
- Oscillation, where a factor such as repeatedly moves between positive and negative values.
- A factor approaching zero multiplied by a bounded function.
- Expressions that can be bounded using inequalities such as .
- Functions whose exact limit is difficult to calculate but whose maximum and minimum behavior is easy to control.
The Key Trigonometric Bound
One of the most important facts for Squeeze Theorem problems is that sine and cosine are always bounded between and :
This simple inequality becomes extremely powerful when you multiply every part by a quantity that approaches zero.
For example, because , multiplying by a positive quantity such as gives:
If both outer expressions approach zero, then the expression containing the sine function must also approach zero.
A Step-by-Step Method
- Look for a bounded component. Common examples are , , or an expression known to stay between two constants.
- Find a lower and upper bound. Use a known inequality to trap the difficult function.
- Make sure the inequality points in the correct direction. If you multiply an inequality by a negative quantity, the inequality signs reverse.
- Take the limit of the lower and upper functions.
- Check whether the two outer limits are equal. If they are, the Squeeze Theorem gives the limit of the middle function.
- State the conclusion clearly. On an AP exam, explicitly identifying the Squeeze Theorem makes your reasoning easy to follow.
Worked Examples
Determine:
Determine:
Determine:
Determine:
Determine:
Why Bounded × Zero Goes to Zero
A very useful pattern to recognize is:
Suppose is bounded. That means there is some constant such that:
If approaches zero, then multiplying the bounded expression by forces the entire product toward zero.
This is exactly why limits such as are approachable with the Squeeze Theorem even though the sine factor itself may oscillate infinitely often.
When the Squeeze Theorem Does Not Work Directly
The Squeeze Theorem requires the two bounding functions to approach the same value. If your bounds approach different values, the theorem does not determine the limit.
For example, if you only know that:
then you have established a bound, but you have not established the limit of . The outside functions must converge to the same number before the Squeeze Theorem can be applied.
Common Mistakes
Expressions involving can oscillate infinitely often near a point. The goal is usually not to figure out what the oscillating factor itself approaches. Instead, bound it and show that the entire expression is forced toward a single value.
Having an upper and lower bound is not enough. The two bounds must approach the same number. Otherwise, the Squeeze Theorem gives you no definite value for the middle function.
If you multiply or divide an inequality by a negative quantity, the inequality signs must reverse. Multiplying by a nonnegative quantity does not reverse them. Always check the sign of the expression you are multiplying by.
Your bounding functions should make the limit easier, not harder. For most AP Calculus problems, simple bounds involving constants, powers of , or familiar functions are enough.
On an AP free-response question, show the limits of the two bounding functions and explicitly state that the Squeeze Theorem applies. The conclusion is the important final step.
A Fast AP Exam Strategy
When you encounter a limit that looks like a Squeeze Theorem problem, ask yourself three questions:
- What part of the function is bounded? Look for sine, cosine, or another expression with a known maximum and minimum.
- What factor is forcing the expression toward a value? Usually this is a power of approaching zero.
- Can I create two simple bounds that have the same limit? If yes, the Squeeze Theorem is probably the intended method.
A useful mental shortcut is:
If the complicated part stays bounded while the other factor approaches zero, the whole product often approaches zero.
Practice Problems
Determine:
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Determine:
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Determine:
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Determine:
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Determine:
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