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Problem 22 Easy Difficulty

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$ \displaystyle \lim_{h \to 0}\frac{(2+h)^5 - 32}{h} $,
$ h = \pm 0.5, \pm 0.1, \pm 0.01, \pm 0.001, \pm 0.0001 $

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Video Transcript

in this problem we are going to guess the value of the limit. If it exists by evil waiting. The function At the given numbers correct. six decimal places Limit. We're gonna guess is limit when age goes to zero of the Kocian do pless age Value to the 5th -32. That's the numerator over age in the denominator room. And the values of age we will be considering our More or less than 0.5 or less than 0.1 Unless your .01 more or less. 0.001 and or less serious .0001. So as we can see we are going to evaluate the function of age given here at points apart. Getting closer to zero, Starting at 7.5 both sides plus a negative 75. We we we start approaching zero because we then take 0.1 which is closer to zero than 0.1 which is even closer and so on. So the function that's a a F of age is two plus age to the faith -32. In that over age. And so we are going to make a table here of human relations of if at these vials of H. So we get each f age and we start writing the results here use a calculator to do that. And we are going to start with negative 0.5 and 0.5. So again -0.50 Point or 88 125. Remember we get to and give the result with correct to the success of all places. And as we get in fact after the six testimony, all the other testimonies are zero. That's your .5. You get one three on 31 you fuck. And we have negative 0.1 and 0.1 at those to get 0.7. Sorry I think I made a mistake here writing the values from my table. Sorry, Start all over again at the at the 3.5548. I meant We got a factor here sorry 48 times point 81 2500. And number 31 point 31 5 serious hero. And they had to 172 point 3 9-1 here in Syria. But you want to get your 88. Sorry I was doing the same mistake here That and this is 88 went 412122 Continue now with 0.01 And 0.01. Oh yeah 0.79 Point 239 33,090 zero then 80 point 80 40 and separating a little bit decimal part. Going to approach to the number. Okay. Year 2.2. Sure. This is a testicle pod 80 40 10 And now we pass to the next group of two points closer to zero that is negative 0.001 And 0.001. Okay, so we get 79 90 00 3.5 14. And then 80 point is built. This tribute here. Okay, 80 point see you will 800 240. And then we have negative 0.0001 two. And then we were all so we get this and this that will be all. So in the negative value we had 79. And I said that we have 992 zero. And And the last one we get 80 points mm 12 zeros aid. And serious again. Mhm. Okay. That's it. You have those results. And we can see that When we get closer to zero at the value of page We get closer and closer to 80. The behavior is that for the negative values of age it is here we have 80 here we have a judge Here. F. Age. So when we come from the left of age 30's Here is zero and when we have negative values we approach We're approaching 80. But this way from this side, so coming from the negative values of age, the function goes to Katie but with values lesson eri mhm. But when we go we come from The right and approaches zero. The function approaches 80. Coming from the Rideau's because you can see here This is greater than 80 And the second degree than 80 is secretary and we are getting closer to 80 but always greater than 80. So this is the behavior but we get easily here. Yes. And the limit when age goes to zero of F. Is equal to pay. And with these exercise taking positive and negative values we are sure that The limit is 80. Uh coming from any size, it's not lateral limit but bi laterally. Okay so now we can explain why this is happening and that's because and we write this limit here to this age Um to the 5th -30-32 is to the 5th that over age. So if we define the function G of X equal X to the 5th. And what we are calculating here is just the limit when age Goes to zero of G. of two Putin H. That's it is we will idea to Placentia footed here and we get Take plus H to the 5th. But we have here -G at two Because these two to the 5th she's 32 that marriage. And we identify this as the definition of the derivative of G two limit but The increment goes to zero if she at a point time plus anti Kermit minus key at the point or the incorrect. That means that if we know how to calculate derivatives in this case very easy. Is the derivative of uh The power of the variable is five times that is five. The sworn in times Same variable to the exponent -1 That is five X to the 4th and if we give away that too, which is what we were calculating with. The limit is equal to five times two To the 4th. That is five times 16, which is 80. So that's the value. We I had to guess for the limit when we do the calculations of the expression F of age, this Kocian here, that corresponds to in the limit when insurers, syrup corresponds to the derivative of this function here, but to compressing.