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Estimating limits from tables

When given a table of values for a function, we can estimate the limit at a certain point by observing the values the function approaches from both sides. The limit is the value the function converges to, even if the function's value at that point is different.

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  • leaf green style avatar for user trash
    My gut says that there must be a function f(x) that gives us this exact table but has the value of 6.37 as its limit. We can't know for sure if the function suddenly (but continuously!) shoots up to 6.37 and back down again or not. So why exactly is it a reasonable estimate to say it doesn't?
    (37 votes)
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    • spunky sam blue style avatar for user Sal Khan
      From the author:I agree with you. We definitely don't know for sure just with a table (which only samples the function). But in this example, we need to pick the most reasonable estimate, and, in general, it is usually most reasonable to assume that something wacky doesn't happen unless it is explicitly pointed out.
      (66 votes)
  • piceratops seed style avatar for user toukaandkaneki12347890
    Can someone explain why 3.68 is a better estimate than 4 for the limit?
    (9 votes)
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  • blobby green style avatar for user Norman Pitts
    It may have saved a lot of confusion if it had been stated in the beginning that the function was not continuous over the interval [4,6].
    (7 votes)
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  • blobby green style avatar for user mthelm85
    3.68 doesn't make sense to me, given the definition of a limit. Since the limit is the value that a function approaches, how could we say that it is equal to 3.68 in this case, since we are no longer approaching that value when x gets really close to 5? The function already takes on a value of 3.68, according to the table, so we must be approaching a value that is different, right?
    (4 votes)
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    • leaf green style avatar for user kubleeka
      We aren't claiming that 3.68 is the limit of the function, only that's that is the most reasonable guess among the given choices. In reality, the limit could be anything at all, or may not exist, but the task at hand is to make a reasonable guess from partial information.
      (6 votes)
  • male robot hal style avatar for user ☣Ƹ̵̡Ӝ̵̨̄Ʒ☢©
    I thought at the limit there would be no value, it would be undefinied. So when the limit goes to 5, and g(x) is 6.37 in this problem, doesn't that mean the limit doesn't exist?
    (2 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      The question reads: What is a reasonable estimate for the limit of 𝑔(𝑥) as 𝑥 approaches 5?

      So, loosely speaking, we are looking for some value 𝐿 (the limit), such that when 𝑥 gets very, very close to 5 (without actually being equal to 5), then 𝑔(𝑥) gets very, very close to 𝐿.

      Looking at the table we see that as 𝑥 goes from 4 to 4.999, 𝑔(𝑥) increases and gets closer and closer to 3.68 .
      And, as 𝑥 goes from 6 to 5.001, 𝑔(𝑥) decreases and gets closer and closer to 3.68 .

      Thereby it is reasonable to assume that the limit of 𝑔(𝑥) as 𝑥 approaches 5 does exist, and that it's value is 3.68 .
      (5 votes)
  • leafers seed style avatar for user Hena Anvar
    Well in n the graph plotted in most of the questions,test,examples has two different points marked 1) A closed (solid) dot means the endpoint is included in the curve and 2)an open dot means it isn't ...My doubt is if it is the open dot then how does the answer here turn out to 3.68
    (2 votes)
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    • leafers seed style avatar for user Ambuj Singh
      Since in question, it is asked "What g(x) approaching as x approaches 5?" When we check for limit from left hand side (i.e., values less than 5) for x = 4.999 (which is very close to 5) we get g(x) = 3.68. Similarly, if we check for limit from right hand side (values that greater than 5), for x = 5.001 we get g(x) = 3.68. So, limit from RHS and LHS are equal, therefore limit for g(x) as x approaches 5 is 3.68.

      It does matter if g(x) is different at x = 5 or in your words there is open dot (or for sake of discussion, lets say undefined) because in question we not asked for value of g(x) at x=5 but what we g(x) is approaching as x approaches 5.
      (5 votes)
  • blobby green style avatar for user veronica62377529
    I really don't get it...So why the answer shouldn't be 6.37??Isn't it the really correct answer for the limit?
    (2 votes)
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    • cacteye yellow style avatar for user Ivy
      The limit at a certain point is the value, that the function seems to be approaching from both sides.
      In this case you can see at minute , that the y-value the function approaches, when x=5, is approximately 3.8.
      Yes, the real value of g(x) at 5 is 6.37, but that isn't what the lines of the function seem to approach.

      Remember, the limit of f(x) at a given point doesn't always equal the value of f(x) at that point, because a limit describes the behavior of functions (what number they are approaching or getting closer and closer to).

      You can also estimate the limit from the table, when x equals 5, g(x) is 6.37, but if you look at the rest of values (from above and from the bottom) you will notice a pattern, for example, from x=4 to x=4.999, the g(x) values get closer and closer to 3.68, the same goes for the values below, from x=6 to x=5.01, the values approach 3.68. And when _x_ equals 5, you'd expect the y-value to be somewhere near 3.8, but no, it falls far apart (6.37), but the tendency of the overall function is to approach (get closer to) 3.68, even thought it never reaches that point.
      (3 votes)
  • piceratops seedling style avatar for user bernardo olisan
    Why using tables instead of graphs for limits? What are the advantages.
    (2 votes)
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  • blobby green style avatar for user 21yuvrajs792
    Do we disregard the solid point on the graph?
    (2 votes)
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  • blobby green style avatar for user ayeshashobnam06
    what would have been the limit if f(4.999) and 5.001 was different numbers rather than both being 3.68.
    4.999 = 3.68
    5.001= 3.69
    (2 votes)
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Video transcript

- [Teacher] The function g is defined over the real numbers. This table gives select values of g. What is a reasonable estimate for the limit is x approaches five of g of x? So pause this video, look at this table. It gives us the x values as we approach five from values less than five and as we approach five from values greater than five it even tells us what g of x is at x equals five. And so, given that, what is a reasonable estimate for this limit? Alright now let's work through this together. So let's think about what g of x seems to be approaching as x approaches five from values less than five. Let's see at four is it 3.37, 4.9? It's a little higher. Is it 3.5, 4.99? Is it 3.66? 4.999; so very close to five. We're only a thousandth away, we're at 3.68. But then at five all of a sudden it looks like we're kind of jumping to 6.37. And once again, I'm making an inference here. I don't, these are just sample points of this function, we don't know exactly what the function is. But then if we approach five from values greater than five. At six we're at 3.97; at 5.1 we're at 3.84. 5.01; 3.7; 5.001; we're at 3.68. So a thousandth below five and a thousandth above five we're at 3.68, but then at five all of sudden we're at 6.37. So my most reasonable estimate would be, well it look like we're approaching 3.68. When we're approaching from values less than five. And we're approaching 3.68 from values as we approach five from values greater than five. It doesn't matter that the value of five is 6.37. The limit would be 3.68 or a reasonable estimate for the limit would be 3.68. And this is probably the most tempting distractor here because if you were to just substitute five; what is g of five? It tells us 6.37, but the limit does not have to be what the actual function equals at that point. Let me draw what this might look like. So an example of this, so if this is five right over here, At the point five the value of my function is 6.37, so let's say that this right over here is 6.37, so that's the value of my function right over there. So 6.37, but as we approach five, so that's four, actually let me spread out a little bit. This obviously is not drawing to scale. But as we approach five, so if this, that's 6.37; then at four, 3.37 is about here and it looks like it's approaching 3.68. So 3.68, actually let me draw that. So 3.68 is gonna be roughly that. 3.68 is gonna be roughly that. So the graph, the graph might look something like this. We can infer it looks like it's doing something like this. Where it's approaching 3.68 from values less than five and values greater than five, but right at five, our value is 6.37. I don't know for sure if this is what the graph look like once again, we're just getting some sample points. But this would be a reasonable inference. And so you can see, our limit. We are approaching 3.68, even though the value of the function is something different.