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## Calculus 1

### Course: Calculus 1>Unit 1

Lesson 9: Squeeze theorem

# Squeeze theorem intro

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.E (LO)
,
LIM‑1.E.2 (EK)
The squeeze (or sandwich) theorem states that if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them. We can use the theorem to find tricky limits like sin(x)/x at x=0, by "squeezing" sin(x)/x between two nicer functions and ​using them to find the limit at x=0. Created by Sal Khan.

## Want to join the conversation?

• I don't actually see the point of this. It's basically because there exist a possibility that f(x)=g(x)=h(x) there for there is one point that all 3 pass through. That's basically it. so is this what the squeeze theorem about?
• Yes on Khan Academy the squeeze theorem is straight forward. In a calc I class you will have to derive the two functions that will squeeze your original function. The squeeze theorem is used on a function where it will be merely impossible to differentiate. Therefore we will derive two functions that we know how to differentiate and we take the derivatives on those two functions at your specific point. Mind you one function has to be greater than or equal to the original function, and the other has to be smaller than or equal to the original function. When you take the derivatives on the two functions and you get the same answer, the original function (since it is between both functions) will also have to be that answer. Hope this helps!
• Around Sal says that the functions don't need to be defined at c only when approaching c? How come this is the case? Don't the functions need to be defined at c as well to properly 'squeeze' the middle function?
• to continue from above, a limit is "getting infinitely close to but not actually dealing with the point".
• what if f(x) and h(x) touch each other on more than 1 point?-
• Then the squeeze theorem applies to all of those points.
• can squeeze theorem only applied on sin or it is applicable on cos and tangent
• You can use squeeze theorem on cosine.
For cosine, you would use -1 <= cosx <= 1 as your starting point.

On tangent, it would be -infinity < tanx < infinity.
If you have tangent, I'd suggest transforming the equation though, so it doesn't involve tangent, but has it in terms of sine and cosine, making it easier to work with.
• How come it's always f(x) or g(x) or h(x) when Sal(and about everybody else) talks about functions? Why can't there be some other names for functions? I never see i(x) or j(x).
• You certainly can use those letters as names for functions. It is just a common practice to start at f and go from there, but you don't have to.
• What will happen if at the point of all the graphs intersection.....both f(x) and h(x) have undefined value.....and have a discontinuity. Is squeeze theorem still applied?
• In general, all derivative operations require the function to be both continuous and differentiable. If either condition is violated, then any related or derived theorems can't be applied.
• Would this only be used when finding the limit of a trig function ?
• It can be used for other functions as well, such as polynomials.
• How can we compare functions, as like in the sandwich theorem f(x)<=g(x)<=h(x).How do we know that f(x) is always less than or equal to h(x)??(Assuming that we cannot draw the graph of the functions manually)
• It is assumed to be true as part of the axioms of the proof.
• Yes, the common point is `(c, f(c))` aka `(c, h(c))` aka `(c, g(c))`. But just as Sal says at , the three functions don't necessarily have to be defined at x = c; only that their limit has to be the same ie., all the three functions tend to the same value (L) at x = c. So this 'common point' might not even be defined.