# A variation of the Sandwich Theorem (Limits) **URL:** \ **Category:** Factual Questions\ **Created:** [January 24, 2018, 4:32pm UTC](https://boards.straightdope.com/t/a-variation-of-the-sandwich-theorem-limits/807386 "2018-01-24T16:32:33Z")\ **Posts on this page:** 9\ **Page:** 1
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [January 24, 2018, 4:32pm UTC](https://boards.straightdope.com/t/a-variation-of-the-sandwich-theorem-limits/807386/1 "2018-01-24T16:32:33Z")
I’ve been thinking about the sandwich theorem and tried to research a variation on it but saw no reference to what I’m am dubbing crossover theorem. My question is is this a known theorem or would I have to prove it? Given f(x) \<= g(x) \<= h(x) for all x \< c f(x) \>= g(x) \>= h(x) for all x \< c lim x-\> c of f(x) = lim x-\> c of h(x) = L Conclusion: lim x-\> c of g(x) = L It seems to me that using the sandwich theorem from the left then the right and showing the two limits are the same would prove the conclusion. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 24, 2018, 4:50pm UTC](https://boards.straightdope.com/t/a-variation-of-the-sandwich-theorem-limits/807386/2 "2018-01-24T16:50:04Z")
> [@](#): > > f(x) \<= g(x) \<= h(x) for all x \< c > f(x) \>= g(x) \>= h(x) for all x \< c I assume that one of those was supposed to say “for all x \> c”? Also, if f and h are continuous, then you don’t need the third line in your givens, as it follows from the Intermediate Value Theorem. It looks pretty straightforward to me: You use the Sandwich Theorem twice, first on the left to prove that the left limit of g(x) = L, and then on the right to prove that the right limit is also L, and since the left and right limits both exist and are equal, that’s the limit. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 24, 2018, 5:01pm UTC](https://boards.straightdope.com/t/a-variation-of-the-sandwich-theorem-limits/807386/3 "2018-01-24T17:01:36Z")
For those who are unfamiliar with the term (as I was before googling), I assume the “sandwich theorem” mentioned here is the same thing I know as the [Squeeze Theorem](https://en.wikipedia.org/wiki/Squeeze_theorem). ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [January 24, 2018, 5:06pm UTC](https://boards.straightdope.com/t/a-variation-of-the-sandwich-theorem-limits/807386/4 "2018-01-24T17:06:05Z")
That just looks like the common Sandwich theorem, or Policemen theorem, or whatever you feel like calling it, except you have renamed the functions on one side, so there is really nothing left to prove here. By the way no extra continuity assumptions are necessary. ETA ok, you need to prove that the renamed functions satisfy the condition in the original theorem, but that is clear after examining each side, as **Chronos** explained. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [January 24, 2018, 5:13pm UTC](https://boards.straightdope.com/t/a-variation-of-the-sandwich-theorem-limits/807386/5 "2018-01-24T17:13:07Z")
Perhaps the term “Sandwich theorem” more often refers to the fact that if you have two pieces of bread and a piece of ham in three-dimensional space, you can always bisect all three with a single planar cut. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 24, 2018, 5:29pm UTC](https://boards.straightdope.com/t/a-variation-of-the-sandwich-theorem-limits/807386/6 "2018-01-24T17:29:52Z")
That’s the Ham Sandwich Theorem. And now that you mention it, I think I learned this theorem as the Squeeze Theorem, but once you get into the body of the OP, it’s clear what he’s asking about, anyway. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [January 24, 2018, 5:32pm UTC](https://boards.straightdope.com/t/a-variation-of-the-sandwich-theorem-limits/807386/7 "2018-01-24T17:32:51Z")
Agree with **Thudlow Boink** , I’ve always known this as the Squeeze Theorem. I think what **Saint Cad** is trying to get at would be the Squeegee Theorem. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [January 24, 2018, 7:30pm UTC](https://boards.straightdope.com/t/a-variation-of-the-sandwich-theorem-limits/807386/8 "2018-01-24T19:30:52Z")
I thought it was a pretty basic application of the squeeze theorem but I was worried that I never saw any reference to the crossover theorem online or in a book. It seems to me to be a lot more practical as functions crossover much more often than are tangent to each other. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 24, 2018, 9:18pm UTC](https://boards.straightdope.com/t/a-variation-of-the-sandwich-theorem-limits/807386/9 "2018-01-24T21:18:19Z")
If I understand correctly, your version would be useful for a situation like the limit as x -\> 0 of [x sin x](http://www.wolframalpha.com/input/?i=y+%3D+x+sin+x,+y+%3D+x,+y+%3D+-x,+x+from+-10+to+10), where g(x) = x sin x is between f(x) = -x and h(x) = x. You could use the “real” Squeeze/Sandwich Theorem with |x| and -|x|, which would be equivalent though perhaps slightly more contrived. And I believe that in any situation where your “crossover theorem” applies, you could translate it via absolute values to a situation where the squeeze theorem directly applies.