if-f-is-continuous-use-lhospitals-rule-to-show-that-displaystyle-lim_hto-0-fracfx-h-fx-h2h-fx-explai

Question

If $ f' $ is continuous, use l'Hospital's Rule to show that
$$ \displaystyle \lim_{h	o 0} \frac{f(x + h) - f(x - h)}{2h} = f'(x) $$
Explain the meaning of this equation with the aid of a diagram.

Amrita Bhasin · Accepted Answer

We know that plugging innuendo put the indeterminate form 00 implying that they must use the hospital&#x27;s rule derivative, the top over derivative of the bottom we must use. Well, Happy tells roll again, cause if we plug him directly, we still won&#x27;t end up with an answer. And then we have to use the law tells you again, because I&#x27;m told when you plug in, you end up with something that isn&#x27;t part of the indeterminate form. You have to keep applying the hospital&#x27;s rule again and again. Okay, this means that the limit as X approaches zero negative co sign to Axe Pole six zero plugging and zero, we end up with negative eight plus six. A equal zero giving us A a cz a is eight over sex, which is the same thing as four over three. And then for the second part, you know what we have. The limit X approaches zero of to crow Son to Axe was three x squared plus B equals zero poking and zero. We end up with two plus B equals zero cause everything else cancels and became B is negative, too. And those are two values

Linh Vu · Answer

because the limit after f x plus march minus f X months arch over that you are. And now watch best jurors There are now if we have the big pitches Oh, we bullet. You hear that? We&#x27;re gonna form the Hamas zero. Therefore we can apply and a little bit and rule. Now then we get the limit on this zero. You give them the denominated could you to really on a topic. Ojeda Graham The express lunch and by the general with him Swan. And now here we minus the a gram of the X minus hunt and by the general in its attempts going to school in here and we send you find this one and then it. Yes. How much does Juzo? We wouldn&#x27;t do this, arch, and we get include you in the program off. Thanks On then, plus a brand X leave any but you there again The Jew m prime on the x, the venue, but you and go to the m Promed X

Amrita Bhasin · Answer

we know that we can rewrite this limit as H A purchase. Zero. That&#x27;s a crime. Expose Age miss off prime X minus age times Negative one, which is of prime of acts, cause extra zeros. Just acts plus f from of acts because X minus zero is also just acts divide by two, which gives us off prime of acts because two F products over two is just one F part of X. Okay, now we know that we can differentiate with respect to H instead of acts we have B is negative, too.

Linh Vu · Answer

it off. Experts lunch honest you f thanks plus half under X minus. So they&#x27;re not distant X. Your Jews their own would have found the, uh this will be the f X minus two f x f x and you go to zero then would you funded by zero. Therefore, we&#x27;re gonna blinding them within rule now and by the Lobaton Roma generally with after just years, there are. And now on the top, we will have no Bram Mamba experts aren&#x27;t and by the general attempts one and then minus here they understood we have zero here. So this time we could hit zero plus the from under X minus front on by the general manager Thames Manasquan here and now on the bottom one handed to us here Way simplified and we get a cogent limit. Under our best user, all every frame on the experts aren&#x27;t. And now, plus that e minor. The cancel X minus are just dividing by too much here, recording from, uh, from grow from 55 we have This will be the definition off the epigram off the X So therefore we have this limit here Equity the F number from off X

Andrew Bassila · Answer

all right, This problem is another proof problem. So they&#x27;ve given us the answer. And the cool thing is, they&#x27;re basically saying, you know, the limit of the difference question equals with limit of the slope of the Sikh in line as X approaches A. So the hint here was the let H equal X minus A. To show that this is indeed true. So we&#x27;re gonna keep the limit as age approaches zero for just for just a moment. What we will change it and just a little bit. But I wanted to address what&#x27;s happening in the actual function for just for to address that first. So if H is equal to X minus a, I should get a plus. And instead of h, I&#x27;m going to write X minus a. And then there&#x27;s no age to substitute over here. And if H is equal to X minus A instead of writing h, I will write X minus a. All right, So what&#x27;s happening in the first bit? Um, more going to continue to drop the right side down a plus X minus. A is just f of X, and then I have of a all over X minus. Hey, so I&#x27;m pretty closed. The only thing I need to do now is it. Toe Show that, you know, if H is approaching zero than X minus, A is going to be approaching zero. Right, And that could be kind of equal to zero or X. Could be a over X could approach A is kind of how you look at it. So now I can write that X approaches A. And I&#x27;ve indeed verified that the limit of the difference quotient could be rewritten as this limit of the slope of a Sikh in line as the two numbers approach each other all right and again, these both worked or both identical pretty much, and this is just one way of writing it against another.

Daniel Jaimes · Answer

this phone number sixty three of the Stuart Calculus eighth edition section two point five prove that if his continues at a if, and only if the limit is six. H approaches zero of the function April&#x27;s Each is equal to that of a and if we were calling a function F is on ly. Continuous is continuous if and only if it&#x27;s limit. As XO. Purchasing of the function is equal to have a mai. So we&#x27;re going to use this. See if we can associate Thies, too. We&#x27;re gonna let h equal explains saying. Ah, and if that&#x27;s true, we can re writes this part and the limit as the limit of the function. April&#x27;s a jewel. A procedure is the same. A saint ex or let me replace it. And that is a plus age which is experiencing. Cols Asperger&#x27;s and then a We can ah keep the same over here. No, since ages explain to say we&#x27;re going to show it as explains a approaching zero. And if we do some more some modifications here a man is a zero, and then here we can add it to both sides of this Aargh. So this would be the same as limited as expert as a or the function f of X because after today so essentially what we have is that at the beginning this is another way of saying the definition of continuity. It is written, written using ancient set of X. Using this conversion, H is equal to x one and saying so we were able to show that this is equivalent to the definition of continuity and thus it is continuous, Ari, for this reason.

Amrita Bhasin · Answer

The problem is, if half crom is continuous use looking task through to show it that limit h close to zero as express age minus X minus stage over to age is a country the derivative of Africa lacks. Explain the meaning of this equation with the aid of their ground. There&#x27;s a limit. Inch goes to zero axp Last image minus half X minus finch off a two page h cost zero most The numerator on didn&#x27;t know Senator Coto zero So we can use of fantastical this&#x27;s culture. Let me to h cost zero from X plus age minus a crime X minus age. I have sniffed you won over to you. They&#x27;re planning a hero too. H we have this week or two from LAX We can draw a graph If this is a function for Max and they have a point X minus age. Thanks On X Plus it, jj half this part is the slope of the thiss second. And what age close to zero? We can see h zero. This is slope of this tender ally at the point of access. But this is true as Prime Alexe

Amrita Bhasin · Answer

Mhm. For this problem. Um Since the limit is in the indeterminate form 0/0, we&#x27;ll use low Patel&#x27;s rule and that&#x27;s going to give us um F of X. Class H minus F of X minus H. All over to H. Um And since we&#x27;re applying the limit as H goes to zero, what we end up getting is when we take the derivative, since we can&#x27;t do that, we&#x27;re going to get F prime have expressed H I&#x27;m 1 uh-. That&#x27;s gonna be, we&#x27;re gonna multiply that by negative one. Eventually it&#x27;ll be plus after prime of X minus H. And this is all gonna be over to. So based on this, when we plug in H equals zero. Now we get two F prime of X over two. So it&#x27;s just going to give us f prime of X. Um as our final answer and therefore we&#x27;ve proven this.

Problem

If $ f" $ is continuous, show that $$ \displayst…

Problem 89 Hard Difficulty

If $ f' $ is continuous, use l'Hospital's Rule to show that
$$ \displaystyle \lim_{h\to 0} \frac{f(x + h) - f(x - h)}{2h} = f'(x) $$
Explain the meaning of this equation with the aid of a diagram.

Answer

$a=\frac{4}{3}$, $b=-2$

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$a=\frac{4}{3}$, $b=-2$

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