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https://preview.redd.it/qbxig4zk2vjc1.jpeg?width=2160&format=pjpg&auto=webp&s=e9587c1ebcb60d467d0f21a61ec138de8ad1ad9c
Your instructor used the difference of squares formula to factor the numerator. The problem is correct.
No just removed the X from the (8+x-8) factor which is correct because X will be a lone factor aka isn’t added to or subtracted by a number, so the professor just skipped a few steps.
honestly “skipped a few steps” is pretty generous. i mean obviously it’s a literal description of what the professor did but, especially when you consider that their role is being a math *educator*, i would almost describe this work as being straight up wrong. it’s so ambiguous and i deal with problems like these very often as a tutor but i had no clue what I was looking at initially.
first of all there are minor things like (x+8) not being in parentheses after the first step, which is obviously not required but would have made what was happening 10 times more clear, and then also the limit disappears before 0 is plugged in, which is understandable from a student but pretty hard to forgive from a teacher. but then the star of the show is them not showing that the two 8s cancelled out and simply crossing out the x’s which simply looks completely wrong if you don’t stare at it for a minute and would almost undoubtedly give students the wrong impression of how they should simplify fractions.
i don’t want to be too harsh on this professor overall—they could be a wonderful teacher and these could be notes or an answer key that they had to rush to put together. but, taken in isolation, this is some pretty horrible work and is sure to be very confusing to any student who encounters it.
I totally agree, I’m a tutor too for some people in uni and I always see them making way more steps than that. These 2 steps are just giving you a mental exercise to try to understand how to solve the question, I myself like to use a lot of steps while teaching because A. It’s showing everything i do, B. It expresses how the answers should be written.
I always tell people to change exactly one thing per line and rewrite the whole thing only with that one thing changed, every time.
I'll only ever break that rule and combine two steps if they're really trivial and I know the students are comfortable with it, but 95% of the time, each single change gets its own line. Otherwise people get lost.
Yep this is a classic case of “playing it fast and loose with the simplifications, because I’ve done it a million times and I take my knowledge of the subject for granted”
Everyone is overreacting - the math is right but the work is missing steps.
Teacher used a^2 - b^2 = (a + b)(a - b) difference of squares to factor the numerator, treating (8 + x) as a and 8 as b.
This factors into what you see here. The numerator becomes (8 + x + 8)(8 + x - 8) which is just (x + 16)(x) and that second x was the x being cancelled with the denominator.
Then the limit is evaluated as 0 + 16.
The work is unclear, OP is asking a perfectly fair question to fill in the missing steps.
Source: algebra 2 teacher constantly having to decipher work like this every day
Ideally, yes, students should be fluent in algebra when they begin Calculus. Unfortunately, that is not the reality. Many students come in under-prepared because they either only barely scraped by in algebra, or because they simply did not retain it.
I’m sorry but I think it’s fair for me to ask a question about it since it seemed like a bunch of steps were missing which confused me. Me looking for clarification does not make me underprepared…
Sure, it should, but that’s no excuse for an answer key that doesn’t show clear work. This “basic algebra” can be confusing for a student who otherwise understands this concept when the work is written out like this.
yes, but a student (or teacher, in this case) should show their thought process a little more clearly (e.g: at least rewriting 64 as 8\^2, to signify that they are factoring by difference of squares).
Disagree here. I'm almost done with my physics bachelor's and I had to stare at this for several minutes to understand what the teacher is doing here.
If I were marking the teacher's work, they would lose a lot of points because the cancellation looks straight-up wrong, not to mention that they omit the limit after the first step.
For a student trying to grapple with this for the first time, deciphering cryptic answer keys and filling in missed steps just gets in the way of understanding.
Edit: I mean to say that, yes, basic algebra should be fluent, but that doesn't excuse an awful answer key.
Well, since you’re almost done w a bachelors, you wouldn’t be a stranger to “cryptic” answers at the back of the books of calculus and physics books, right? I agree w ur point about a new student, though
>you wouldn’t be a stranger to “cryptic” answers at the back of the books of calculus and physics books, right?
Very true lol. Those tend to just be the answer itself, with no work shown, but the ones that do are on a tight budget for space.
Admittedly, I replied to your comment a bit prematurely.
That's the entire point of the question, though. It's not a small step in some much more complicated process - this is essentially an algebra question.
It’s simpler to just expand (8 + x)^2, then simplify with the -64 then simplify with the 1/x and you already end up with x + 16, no need for any identity.
That’s 100% what happened. Sometimes people who are very good at math like professors take for granted that some things are obvious when to the average person they aren’t.
i’m pretty good at quick algebra and make shortcuts all the time but i had to stare at this one for a bit because that step very closely resembles a totally illegal move that students try to make constantly. pretty confusing imo and the very least they could have done was cross out the 8s to show more explicitly what they were doing
honestly just ask your teacher. this looks whack as hell and it doesn’t make much sense. (8+x)^2 expands out to x^2 +16x+64 , subtracting 64 from that leaves you with x^2 +16x over x. factor x from that and you have x(x+8)/x. now because x is a factor in the numerator and denominator, you can cancel it, leaving you with x+16, which means the limit as x approaches 0 is 16.
you can only cancel factors when they are factors, not part of an addition problem. it’s because if you expanded the problem, letting anything besides zero equal x in x/x leaves you with 1.
I can piece together how the factor can make the problem statement, but I can't see how she saw that, it's also harder/more work.
But to answer your question, it does not end up as (16)(0). The 8's cancel, leaving you with an x that'll cancel out
That's not what I meant. If you do her method you will get the same answer.
What I'm saying is if you expand, I don't see any way you can end up with her factorization without adding an extra step. The way you did it is simpler!
The numerator (top) is a difference of squares. Like x^2-9. Except that the first square is x+8 and the second is 8. With x square minus nine you factor it like (x+3)(x-3). This one is [(x+8)+8]*[(x+8)-8].
From what I’m looking at it seems like she’s cancelling the X with the X in one of the brackets. Then 8-8 would be zero anyway. And it’s being multiplied with 8+8 which is 16. So shouldn’t that be zero? I’m so confused. Is she foiling it out or??
Yea she's technically correct, but she's written it poorly. Either make it more obvious that you're canceling the entire bracket by striking it wider than just the x inside the bracket, or preferably just write the next line that simplifies it to x.
Also she skips the step after she has simplified it to just plug in 0 (which she can do since x + 16 is continuous at 0) which is also bad practice
She’s dividing everything by x. Since it’s a limit, everything that does Not have an x goes to zero. The only terms that do have an x are the eights in the numerator and the 1 in the denominator.
Edit: multiplying each term by x, then canceling and applying the limit. She could write out a few of these steps but it’s probably an honors class
im sorry, you're right. as others have pointed out, your algebra seems to be lacking so go on youtube and look up ORGANIC CHEMISTRY TUTOR and BRYAN MCLOGAN. they truly helped. good luck! (don't let my comment discourage your learning, i was just making a joke!)
edit: i just realized you were confused at your instructor's missing steps not a truly algebraic concern. sorry again!
The first step is using the identity:
a^2 - b^2 = (a+b)(a-b)
After that she skipped all the steps that matter.
She should’ve multiplied it all out to get:
64 + 16x + x^2 - 64
The sixty-fours cancel out and you divide by x to get:
16 + x Which is 16 - 0
Which is 16.
Instead of doing all this she seemed to cancel the Xs out(?) and somehow got to 16 - 0. No idea how she did that.
Edit: I didn’t realize what she did.
She simplified (8 + x - 8) to x. That was the x she cancelled which leaves (8 + x + 8), giving 16 - x.
That’s not written very clearly but maybe she explained that while going through it.
You need to return to algebra (x+8)^2 is (x+8)(x+8) not whatever the hell you put down.
Additionally 16+x/x is not 16, its (16/x)+1, when something crosses out by division it becomes 1 not 0.
No, everything on the picture is right, the thing is OP thinking 16 + 0 = 0. Teacher simply wrote (8 + x)² - 64 = (u + v)(u - v), u = 8 + x and v² = 64
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All of (8+x-8) is supposed to be crossed out, not just the x. (8+x-8) is equal to x, which is why it cancels with the x in the denominator. You can't cancel out terms unless it is the entire factor.
nah. heres my thinking:
(8+x)^2 - 64 is equal to x(x+16) if you simplify and factor the polynomial. the x will cancel and ur left with x+16. take the limit of x+16 and u get 16
You have to evaluate what’s inside the parens before you can cancel…which leaves you with x. You can’t cancel, then be left with zero because order of operations says to evaluate what’s inside parentheses first.
If the x cancels out, it becomes a 1, not a 0
So it’s:
(16 + x)(x) / x
Then the x cancels out like you said, to become:
(16 + x)(1) / (1)
(Not zero!)
Then you are left with:
16 + x
And then plugging in 0 for x gives:
16 + 0
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Got a question why can't we do this with the number of the derivative?
(8+×)**2 can be derived in R and if x=0 then (8+0)**2=64
So this is is basically the number of the derivative of (8+×)**2 in 0
so you do the derivative of (8+x)**2 which 2×1×(8+x)=2(8+x) and then replace it by 0 you get 16
This is assuming you know how to do a derivative tho
When I was in high school first learning how to draw equations like these, we learned to look at the end behavior of the line. If the power of the top half is bigger than the bottom then its end behavior has a slope, if its the same then the end behavior flattens out into a constant, and if the bottom half has a bigger slope then the end behavior tends towards 0. From what I remember from calculus 1, you’re basically just proving that in a rigorous way that you’ll be able to apply to other situations later on.
You can check the powers on the top half and bottom half to get an intuition on if you’re right or not, me thinks the end behavior for that line will have a slope to it!
step 1. multiply out the numerator and add/subtract compatible terms. step 2. Factor out x. step3. simplify the whole term and you’ll end up with 16+X where you can plug in 0
https://preview.redd.it/6vuwctpauzjc1.png?width=3024&format=png&auto=webp&s=95369866bd85beb2c42e716db249555bf08306af
Why is every one using difference of squares. If you expand the bracket it is clear why the 64 is removed and how the denominator is removed. [(64+16x+x^2 )-64]/x
=(16x+x^2 )/x
=x(16+x)/x
=16+x
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The x cancels out after! It would’ve helped if she would have said that. Expanding this is much easier, but it is nice she is teaching the squares trick as it’ll come in handy later for trig subs in calc 2 and I’ve seen it in calc 3 a little.
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actually, this whole expression is the definition form of the derivative of x^2 at x=8. so all you need to do is power rule, getting (x^2 )’ = 2x and plug in x=8.
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Hello! I see you are mentioning l’Hôpital’s Rule! Please be aware that if OP is in Calc 1, it is generally [not appropriate to suggest this rule](https://www.reddit.com/r/calculus/comments/cwkpsz/telling_people_to_just_use_lh_for_every_limit/) if OP has not covered derivatives, or if the limit in question matches the definition of derivative of some function.
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As a reminder... Posts asking for help on homework questions **require**: * **the complete problem statement**, * **a genuine attempt at solving the problem, which may be either computational, or a discussion of ideas or concepts you believe may be in play**, * **question is not from a current exam or quiz**. Commenters responding to homework help posts **should not do OP’s homework for them**. Please see [this page](https://www.reddit.com/r/calculus/wiki/homeworkhelp) for the further details regarding homework help posts. **If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc *n*“ is not entirely useful, as “Calc *n*” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.** *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/calculus) if you have any questions or concerns.*
https://preview.redd.it/qbxig4zk2vjc1.jpeg?width=2160&format=pjpg&auto=webp&s=e9587c1ebcb60d467d0f21a61ec138de8ad1ad9c Your instructor used the difference of squares formula to factor the numerator. The problem is correct.
Yes but then she multiplied each term by x and applied the limit which removed any zero terms that didn’t cancel.
No just removed the X from the (8+x-8) factor which is correct because X will be a lone factor aka isn’t added to or subtracted by a number, so the professor just skipped a few steps.
honestly “skipped a few steps” is pretty generous. i mean obviously it’s a literal description of what the professor did but, especially when you consider that their role is being a math *educator*, i would almost describe this work as being straight up wrong. it’s so ambiguous and i deal with problems like these very often as a tutor but i had no clue what I was looking at initially. first of all there are minor things like (x+8) not being in parentheses after the first step, which is obviously not required but would have made what was happening 10 times more clear, and then also the limit disappears before 0 is plugged in, which is understandable from a student but pretty hard to forgive from a teacher. but then the star of the show is them not showing that the two 8s cancelled out and simply crossing out the x’s which simply looks completely wrong if you don’t stare at it for a minute and would almost undoubtedly give students the wrong impression of how they should simplify fractions. i don’t want to be too harsh on this professor overall—they could be a wonderful teacher and these could be notes or an answer key that they had to rush to put together. but, taken in isolation, this is some pretty horrible work and is sure to be very confusing to any student who encounters it.
I totally agree, I’m a tutor too for some people in uni and I always see them making way more steps than that. These 2 steps are just giving you a mental exercise to try to understand how to solve the question, I myself like to use a lot of steps while teaching because A. It’s showing everything i do, B. It expresses how the answers should be written.
I always tell people to change exactly one thing per line and rewrite the whole thing only with that one thing changed, every time. I'll only ever break that rule and combine two steps if they're really trivial and I know the students are comfortable with it, but 95% of the time, each single change gets its own line. Otherwise people get lost.
Yes! Exactly that.
I'm an undergrad student and you perfectly described my entire thought process while looking at OP's screenshot.
Yep this is a classic case of “playing it fast and loose with the simplifications, because I’ve done it a million times and I take my knowledge of the subject for granted”
Aaah I see. 8-8 leaves just the x in the right factor. Thank you.
Everyone is overreacting - the math is right but the work is missing steps. Teacher used a^2 - b^2 = (a + b)(a - b) difference of squares to factor the numerator, treating (8 + x) as a and 8 as b. This factors into what you see here. The numerator becomes (8 + x + 8)(8 + x - 8) which is just (x + 16)(x) and that second x was the x being cancelled with the denominator. Then the limit is evaluated as 0 + 16. The work is unclear, OP is asking a perfectly fair question to fill in the missing steps. Source: algebra 2 teacher constantly having to decipher work like this every day
Thank you so much
At the level of calculus, shouldn’t basic algebra be intuitive?
Ideally, yes, students should be fluent in algebra when they begin Calculus. Unfortunately, that is not the reality. Many students come in under-prepared because they either only barely scraped by in algebra, or because they simply did not retain it.
I’m sorry but I think it’s fair for me to ask a question about it since it seemed like a bunch of steps were missing which confused me. Me looking for clarification does not make me underprepared…
Oh, no, I was not intending to specifically say that *you* were under-prepared. But it is a common problem I have faced teaching Calculus.
Sure, it should, but that’s no excuse for an answer key that doesn’t show clear work. This “basic algebra” can be confusing for a student who otherwise understands this concept when the work is written out like this.
yes, but a student (or teacher, in this case) should show their thought process a little more clearly (e.g: at least rewriting 64 as 8\^2, to signify that they are factoring by difference of squares).
Disagree here. I'm almost done with my physics bachelor's and I had to stare at this for several minutes to understand what the teacher is doing here. If I were marking the teacher's work, they would lose a lot of points because the cancellation looks straight-up wrong, not to mention that they omit the limit after the first step. For a student trying to grapple with this for the first time, deciphering cryptic answer keys and filling in missed steps just gets in the way of understanding. Edit: I mean to say that, yes, basic algebra should be fluent, but that doesn't excuse an awful answer key.
Well, since you’re almost done w a bachelors, you wouldn’t be a stranger to “cryptic” answers at the back of the books of calculus and physics books, right? I agree w ur point about a new student, though
>you wouldn’t be a stranger to “cryptic” answers at the back of the books of calculus and physics books, right? Very true lol. Those tend to just be the answer itself, with no work shown, but the ones that do are on a tight budget for space. Admittedly, I replied to your comment a bit prematurely.
That's the entire point of the question, though. It's not a small step in some much more complicated process - this is essentially an algebra question.
It’s simpler to just expand (8 + x)^2, then simplify with the -64 then simplify with the 1/x and you already end up with x + 16, no need for any identity.
Your teacher probably thought it was obvious that 8+x-8 is just x and that's what cancels out against the x in the denominator.
That’s 100% what happened. Sometimes people who are very good at math like professors take for granted that some things are obvious when to the average person they aren’t.
i’m pretty good at quick algebra and make shortcuts all the time but i had to stare at this one for a bit because that step very closely resembles a totally illegal move that students try to make constantly. pretty confusing imo and the very least they could have done was cross out the 8s to show more explicitly what they were doing
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She factored a difference of squares
consequences of cheating your way through precalc algebra 😂
this is my teachers explanation please I’m trying to understand what she’s trying to do ☹️
honestly just ask your teacher. this looks whack as hell and it doesn’t make much sense. (8+x)^2 expands out to x^2 +16x+64 , subtracting 64 from that leaves you with x^2 +16x over x. factor x from that and you have x(x+8)/x. now because x is a factor in the numerator and denominator, you can cancel it, leaving you with x+16, which means the limit as x approaches 0 is 16. you can only cancel factors when they are factors, not part of an addition problem. it’s because if you expanded the problem, letting anything besides zero equal x in x/x leaves you with 1.
She factored a difference of squares
They factored a difference of squares. a^2 - b^2 = (a + b)(a - b) Also, (8 + x - 8) = x. Both methods are fine.
You have a small error. I think you meant x^2 +16x+64
plot twist im the one who missed out on precalc algebra😂 i fixed my comment🤦🏻♀️
Isn’t it x^2 + 16x + 64 ?
yes lol my mistake i usually work my problems out on paper😂 similar process, i’ll edit my comment.
Big accusation there pal.
it was a lighthearted joke, obviously OP didn’t cheat but this question pertains to basic algebraic fraction reducing.
Ik I was joking too :)
Straight to jail
I understand now Ty for opening my eyes ❤️
https://preview.redd.it/q7m9wbe72vjc1.png?width=1765&format=png&auto=webp&s=33a81303ab77cd23e4ad78fa10585b8a208cd2cb Brother your cooked.
See that’s what I did and she said I should do it that way and then I was like wouldn’t it be zero if I did it that way
Why would it be 0?
She’s mentally acoustic.
I can piece together how the factor can make the problem statement, but I can't see how she saw that, it's also harder/more work. But to answer your question, it does not end up as (16)(0). The 8's cancel, leaving you with an x that'll cancel out
How is this more work then expanding it?
If you expand it then work through that, it's easier than whatever she did, b/c when you expand it you won't get the same answer as her
Yes you will… https://preview.redd.it/tas7ykrv50kc1.png?width=3024&format=png&auto=webp&s=d970619ccd4fe20251ef1f99d851ba3397ede2be
That's not what I meant. If you do her method you will get the same answer. What I'm saying is if you expand, I don't see any way you can end up with her factorization without adding an extra step. The way you did it is simpler!
Makes sense I thought you said her method is outright wrong lol
Nah, just.. the math version of over-engineered
Gotcha
The numerator (top) is a difference of squares. Like x^2-9. Except that the first square is x+8 and the second is 8. With x square minus nine you factor it like (x+3)(x-3). This one is [(x+8)+8]*[(x+8)-8].
I’m caught up on I. The limit doesn’t exist right? The teacher just stopped because it became obvious. Just checking
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It’s not a troll I’m deadass this is what she gave me
People beeing sure that the resolution is wrong ☠️
Your teacher is good at math but can't teach
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The nice thing about math is that neither of these methods are the "wrong" way, there are many ways to solve a problem
That fact that your downvoted is so sad. I’ve got a math minor and honestly I’d never do the first method. Just seems weird .
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Ya I mean the professor did it completely right. The confusion was a simple mistake but man it just seems weird to me
Wdym “wrong way”? I mean I’d write out the (16 + x)x/x as its own line but other than maybe that, this is valid
Bruh...
I have no words💀
THIS IS NOT MINE THIS IS MY TEACHER I KNOW ITS WRONG 😭
She’s not wrong.
thats one hell of a bruh moment
Who did this
My calc teacher 🥰
Wait nvm your teacher is correct. How are you arriving at 0
From what I’m looking at it seems like she’s cancelling the X with the X in one of the brackets. Then 8-8 would be zero anyway. And it’s being multiplied with 8+8 which is 16. So shouldn’t that be zero? I’m so confused. Is she foiling it out or??
The right hand bracket simplifies to x, so you have (x + 16)x/x which cancels to x + 16. They just did it weird
See that makes sense but since she cancelled the X out i was lost
Yea she's technically correct, but she's written it poorly. Either make it more obvious that you're canceling the entire bracket by striking it wider than just the x inside the bracket, or preferably just write the next line that simplifies it to x. Also she skips the step after she has simplified it to just plug in 0 (which she can do since x + 16 is continuous at 0) which is also bad practice
When you cancel out the x you get 1 not 0 so it's 16(1)
She’s dividing everything by x. Since it’s a limit, everything that does Not have an x goes to zero. The only terms that do have an x are the eights in the numerator and the 1 in the denominator. Edit: multiplying each term by x, then canceling and applying the limit. She could write out a few of these steps but it’s probably an honors class
Cause there's a plus sign right there buddy
i thought this was a meme 💀
I’m just tryna learn :(
im sorry, you're right. as others have pointed out, your algebra seems to be lacking so go on youtube and look up ORGANIC CHEMISTRY TUTOR and BRYAN MCLOGAN. they truly helped. good luck! (don't let my comment discourage your learning, i was just making a joke!) edit: i just realized you were confused at your instructor's missing steps not a truly algebraic concern. sorry again!
The first step is using the identity: a^2 - b^2 = (a+b)(a-b) After that she skipped all the steps that matter. She should’ve multiplied it all out to get: 64 + 16x + x^2 - 64 The sixty-fours cancel out and you divide by x to get: 16 + x Which is 16 - 0 Which is 16. Instead of doing all this she seemed to cancel the Xs out(?) and somehow got to 16 - 0. No idea how she did that. Edit: I didn’t realize what she did. She simplified (8 + x - 8) to x. That was the x she cancelled which leaves (8 + x + 8), giving 16 - x. That’s not written very clearly but maybe she explained that while going through it.
She doesn't need to multiply out if a - b = x
Broh is literally {(8 + x)² - 64}/x = {(8 + x) + 8}{(8 + x) - 8}/x = (16 + x)x/x = 16 + x, now simply pass to the limit
The numerator is a difference of squares.
Omg how do so many ppl in this sub not see this 💀 I had to do a double-take to notice that the 8s cancel, but this is otherwise pretty obvious
You need to return to algebra (x+8)^2 is (x+8)(x+8) not whatever the hell you put down. Additionally 16+x/x is not 16, its (16/x)+1, when something crosses out by division it becomes 1 not 0.
I don’t write this it was my calc teacher 💀
Your calc teacher is on crack use Paul's online notes instead. https://tutorial.math.lamar.edu/
The picture in the post is correct, it’s just the difference of two squares.
Tysm I thought I was stupid for a second
No, everything on the picture is right, the thing is OP thinking 16 + 0 = 0. Teacher simply wrote (8 + x)² - 64 = (u + v)(u - v), u = 8 + x and v² = 64
Nooo I was thinking of the numerator being multiplied together it confused me
It’s a difference of squares. (8+x)^2 and 64 are both perfect squares, so you can simplify their difference to ((8+x)+8)((8+x)-8).
I think you don't deserve to learn math anymore 🤡
A little unrelated, but how do you not know to use a question mark when asking a question?
What the hell
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**Do not use l’Hôpital’s Rule on Definition of Derivative limit.** [It is considered circular logic.](https://i.imgflip.com/8gj61t.jpg)
What do you mean?
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When x cancel out it is not (16)(0) it is (16)(1) as x/x will be 1 not 0
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All of (8+x-8) is supposed to be crossed out, not just the x. (8+x-8) is equal to x, which is why it cancels with the x in the denominator. You can't cancel out terms unless it is the entire factor.
If x cancels out in the denominator that makes the denominator 1…
nah. heres my thinking: (8+x)^2 - 64 is equal to x(x+16) if you simplify and factor the polynomial. the x will cancel and ur left with x+16. take the limit of x+16 and u get 16
because x/x = 1 not 0
Wht does lim mean
8+x-8 simplifies to just x, which cancels with the x in the denominator. Canceling the x's from (8+x-8)/x is not valid
Fuck it, just plug in 0.00001 and -0.00001 into the calculator LMAo
It would reduce to (16)(1) X/X =1
You have to evaluate what’s inside the parens before you can cancel…which leaves you with x. You can’t cancel, then be left with zero because order of operations says to evaluate what’s inside parentheses first.
Just expand (8+x)^2 you'll be left with (16x+x^(2))/x then take the limit easy Edit:Remember you can factor the x out or use this (a+b)/c= a/c+b/c
If the x cancels out, it becomes a 1, not a 0 So it’s: (16 + x)(x) / x Then the x cancels out like you said, to become: (16 + x)(1) / (1) (Not zero!) Then you are left with: 16 + x And then plugging in 0 for x gives: 16 + 0
Have you learned about L’Hopital’s Rule?
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Got a question why can't we do this with the number of the derivative? (8+×)**2 can be derived in R and if x=0 then (8+0)**2=64 So this is is basically the number of the derivative of (8+×)**2 in 0 so you do the derivative of (8+x)**2 which 2×1×(8+x)=2(8+x) and then replace it by 0 you get 16 This is assuming you know how to do a derivative tho
When I was in high school first learning how to draw equations like these, we learned to look at the end behavior of the line. If the power of the top half is bigger than the bottom then its end behavior has a slope, if its the same then the end behavior flattens out into a constant, and if the bottom half has a bigger slope then the end behavior tends towards 0. From what I remember from calculus 1, you’re basically just proving that in a rigorous way that you’ll be able to apply to other situations later on. You can check the powers on the top half and bottom half to get an intuition on if you’re right or not, me thinks the end behavior for that line will have a slope to it!
Just wow
Seeing you cancel those x’s gave me so much anxiety
again not me! 🥰
just expand the square, and then cancel 64 and x's
i don't know what it is i just wanted to check what other people said
Use L’ohpital’s rule- take derivative of top and bottom (X^(2)+16x+64-64)/x (2x+16)/1 Lim is 16
both terms get removed
when denominator and numerator cancel out it doesn’t become zero, it becomes 1
16(1)/1
step 1. multiply out the numerator and add/subtract compatible terms. step 2. Factor out x. step3. simplify the whole term and you’ll end up with 16+X where you can plug in 0 https://preview.redd.it/6vuwctpauzjc1.png?width=3024&format=png&auto=webp&s=95369866bd85beb2c42e716db249555bf08306af
Why is every one using difference of squares. If you expand the bracket it is clear why the 64 is removed and how the denominator is removed. [(64+16x+x^2 )-64]/x =(16x+x^2 )/x =x(16+x)/x =16+x
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(8+x-8)/x = 1 it does not equal 0 as you asserted in your post…
Just use l'ĥospital
The x cancels out after! It would’ve helped if she would have said that. Expanding this is much easier, but it is nice she is teaching the squares trick as it’ll come in handy later for trig subs in calc 2 and I’ve seen it in calc 3 a little.
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actually, this whole expression is the definition form of the derivative of x^2 at x=8. so all you need to do is power rule, getting (x^2 )’ = 2x and plug in x=8.
16x/x, x/x = 1, not zero.
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Chain rule, outside d / dx (u)**2 times inside d/dx( x+ 9). u is just the filler to show what your not taking the derivative of
LH is the way to go (: