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Squeeze theorem example

An example applying the squeeze theorem. Created by Sal Khan.

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  • male robot johnny style avatar for user Sidhusan Devamanoharan
    At , when sal takes the limits , he uses the (greater or equal to inequality) sign. But shouldnt it be actually only equal sign , as the limits are equal as x approaches 2.
    (43 votes)
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    • blobby green style avatar for user Creeksider
      The limits are in fact equal, and it's easy enough to see that without resort to the squeeze theorem. The point of this exercise, though, is to show how the squeeze theorem could be used to establish this limit, so we use the inequality until the final step.
      (32 votes)
  • mr pants teal style avatar for user Dorian Thiessen
    How do we know when we take the principal root (the positive root) or when it's the square root (negative and positive roots)?
    (11 votes)
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  • piceratops ultimate style avatar for user Matěj Krátký
    I just don't see any point in using the squeeze theorem. The limit could be easily found by factoring the expression in the numerator.
    (4 votes)
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  • piceratops seedling style avatar for user Yasir Mehdi
    Up till now i have watched almost every video of "limits" and the idea is pretty clear but what keeps me wondering is whether sal would talk about other ways of solving the limit functions rather then just using algebra or rationalising the functions. In school i was taught to differentiate the numerator and denominator of the function and put the value of x in it . For eg we need to find the limit of x^2-3x+2/x-2 ...after differentiating numerator and denominator we are getting 2x-3/1 , and by just putting the value of x in the function to get 1 . Why do we really need to differentiate ?
    (2 votes)
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    • leafers ultimate style avatar for user The Chosen One
      Your teachers were demonstrating a theorem called L' Hopitals Rule. In that case it might have not been so helpful but in other cases if you just plug it in you might get 0/0. In that case L'Hopital claimed that taking the limit of the derivative of the numerator/ the derivative of the denominator would result in the same limit.
      (6 votes)
  • leafers ultimate style avatar for user ben
    Wouldn't x confuse you with times?
    (2 votes)
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    • aqualine seed style avatar for user Zeta
      Nope. It is a widespread convention to write times as a point, or as implicitly there when there is nothing. I mean when for example you see (5x+6y+4z); this equals [ 5*x+6*y+4*z]. Wouldn't it become pretty tricky to write these kinds of formulae with x's and y's, if we write the product with an x?
      (5 votes)
  • piceratops ultimate style avatar for user Robin Carlier
    isn't calculate the limit using algebra faster? Because with this theorem we first have to find the two functions that "sandwich" our function, then find proof that they are are indeed greater/less or equal than our function. Then we can really use the theorem. With algebra we can find the limit much faster
    (3 votes)
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    • piceratops ultimate style avatar for user Just Keith
      You would normally use this theorem for those instances of a limit of which you don't know how to find the limit or for which the limit is too impractical to compute. So, you sandwich the difficult limit with two relatively easy limits.

      So, if we could readily find the limit of the function, then we wouldn't bother with the squeeze theorem. So, it is ordinarily just used for the really hard to find limits.
      (2 votes)
  • duskpin ultimate style avatar for user Maegan Johnson
    When is this practical? I'm not challenging it, I believe it is, I'm just struggling to grasp when it can be used other than math-class-problems. When/how can you have a function, and easily know that it's less than or equal to some other function and greater than or equal to a second function? Especially without graphs to look at?
    (1 vote)
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    • leaf green style avatar for user jude4A
      It is used primarily in the world of mathematics to help us understand the behavior of certain functions we didn't fully understand by using other functions which we did understand. In regards to their inequality, you can compare them by comparing inputs and outputs of the functions. Using your results, you could see which is greater.
      (5 votes)
  • starky ultimate style avatar for user Sirsho
    If the three plotted fuctions meet at more than one point keeping the condition of squeeze theorem valid, periodically after a certain interval, then what to do?
    (2 votes)
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  • blobby green style avatar for user Arghadeep Modak
    i have got the point when x approaches to 2 and we have the squeeze theorem.but what happens when we have x approaches to 3 ? then there is no squeeze happening. we have just a ineqality .
    (1 vote)
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    • aqualine ultimate style avatar for user Mark Geary
      I thought this video was pretty clear. At each value of x, the functions f, g, an h are in order of magnitude: f (x) <= g (x) <= h (x). So, at x = 3, g is between f and h. As we approach x = 2, the functions all converge, and g is driven to the value of 1, between f's value of 1 and h's value of 1.
      (2 votes)
  • blobby green style avatar for user Tay Tay
    also you could turn the second function into a form where you'd be able to plug in the 2 without making it indeterminate by factoring the top right?
    (1 vote)
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Video transcript

The graphs of f of x, g of x, and h of x are shown below. Select and drag cards to create a compound inequality that orders the values of f of x, g of x, and h of x for x-values near 2 but not at 2 itself. So for any of the x-values that are depicted right over here, say, x is equal to 3, we see that h of 3 is the largest, f of 3 is the smallest, and g of 3 is in between. And that's true for any of the x-values that they've depicted here. If we look at when x is equal to 1, h of 1 is the largest, f of 1 is the smallest, g of 1 is in between. So for all of the x-values that they're depicting, f of x is less than or equal to g of x, which is less than or equal to h of x. And the only place where they equal based on this graph comes into play, it looks like as we approach x equals 2, it looks like all the functions are approaching 1. So that's where the equal might come into play. But let's keep seeing what they want us to do after that. So then it says it follows that. And instead of writing f of x, g of x, and h of x, they've written the actual definitions of them. So let's just remind ourselves f of x is 2x times the square root of x minus 1 minus 1. That's this blue one right over here. So instead writing f of x, we can write 2 times the square root of x minus 1. Minus 1 is less than or equal to g of x. G of x was this rational expression right over here. So let's go back down here. We get this rational expression. And then that's going to be less than or equal to h of x, which was, I believe, e to the x minus 2. Is that right? Yep, e to the x minus 2. So all we've really done is replaced f, g, and h with their definitions. And then this means that the limit-- so they're looking at the limit as x approaches 2 of these different expressions. So the limit as x approaches 2 of this expression is going to be less than or equal to the limit as x approaches 2 of this expression, which is this right over here, which is going to be less than or equal to the limit as x approaches 2 of this expression, which is that right over there. And then they say finally the value of the limit as x approaches 2 of this thing right over here is-- well, this is where the squeeze theorem comes into play. We just have to remind ourselves-- well, let's just think about it. Can we figure out the limit as x approaches 2 of this right over here? Well, the limit as x approaches 2-- let's see, 2 minus 1. So we're taking the principal root of 2 minus 1, which is the principal root of 1. So you have 2 times 1 minus 1. So this is 1. This right over here is e to the 2 minus 2. That's either the 0, or that's 1 as well. So the limit of all of this is going to be greater than or equal to 1, and it's going to be less than or equal to 1. Or it's right in between 1 and 1. And the only way that it's going to be between 1 and 1 is if it is equal to 1. This is the squeeze theorem at play right over here. g of x, over the domain that we've been looking at, or over the x-values that we care about-- g of x was less than or equal to h of x, which was-- or f of x was less than or equal to g of x, which was less than or equal to h of x. And then we took the limit for all of them as x approached 2. For the lower function, for f of x, it approached 1. And we see it in the graph right over here. In the lower function, f of x approaches 1, h of x approaches 1, and therefore, g of x must also approach 1. And we actually see that in this graph right over here. But anyway, we could check our answer just to feel good about ourselves. We got it right.