AP Calculus BC Flashcards: Determining Limits Using the Squeeze Theorem
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What is the primary purpose of the Squeeze Theorem in calculus?
The Squeeze Theorem is a method used to determine the limit of a function when other methods, like direct substitution or algebraic manipulation, are difficult to apply.
Card 1 of 10
All Flashcards (10)
What is the primary purpose of the Squeeze Theorem in calculus?
The Squeeze Theorem is a method used to determine the limit of a function when other methods, like direct substitution or algebraic manipulation, are difficult to apply.
If you know that 2x ≤ f(x) ≤ x² + 1 for all x, can you use the Squeeze Theorem to find the limit of f(x) as x approaches 1?
Yes, because as x approaches 1, the limit of 2x is 2 and the limit of x² + 1 is also 2, so the limit of f(x) must be 2.
What is the fundamental condition required to apply the Squeeze Theorem to a function f(x)?
The function f(x) must be 'squeezed' or bounded between two other functions, g(x) and h(x), such that g(x) ≤ f(x) ≤ h(x) over an interval around the point of interest.
Why is the Squeeze Theorem often called the 'Sandwich Theorem' or 'Pinching Theorem'?
It is called this because the function of interest is metaphorically sandwiched or pinched between two other functions that converge to the same value.
A function f(x) is trapped between y = cos(x) and y = -cos(x) near x = π/2. Can the Squeeze Theorem be used to find the limit of f(x) as x approaches π/2?
Yes, since both cos(x) and -cos(x) approach a limit of 0 as x approaches π/2, the Squeeze Theorem can be used to determine the limit of f(x) is also 0.
If g(x) ≤ f(x) ≤ h(x), and you know that lim(x→c) g(x) = L and lim(x→c) h(x) = L, what can you conclude about the limit of f(x)?
Using the Squeeze Theorem, you can conclude that the limit of f(x) as x approaches c must also be L.
Define the Squeeze Theorem.
The Squeeze Theorem states that if a function is bounded between two other functions that share a common limit at a certain point, then the function itself must also have that same limit at that point.
What are the two general methods mentioned for determining the limits of functions?
The two methods mentioned are using equivalent expressions for the function and using the Squeeze Theorem.
In the Squeeze Theorem, what must be true about the limits of the two outer (bounding) functions?
For the theorem to be conclusive, the limits of the two bounding functions must be equal to each other as x approaches the specified point.
What is the key takeaway from the Squeeze Theorem regarding a function's behavior?
The theorem shows that the limit of a function can be determined indirectly by understanding the behavior of functions that bound it from above and below.