imals. So how will it ever reach $b$? Do I have any misconceptions regarding calculus and infinitesimals?

A:

In his book Infinite Powers, Steven Strogatz mentions this exact issue which you are describing. He considers the difference between 'completed infinity' and 'potential infinity'. I will get to your particular issue soon, but first, I will describe a similar problem that helps  pinpoint what is going on.
As you know, division by zero is not allowed in mathematics. However, many people share in the false belief that
$$
1/0 = \infty
$$
Why is this? A trained mathematician might revile at the above statement, but actually, it is very plausible; wrong, but plausible.  Take a look at this:
\begin{align}
1/0.1&=10 \\
1/0.01&=100 \\
1/0.001&=1000 \\
1/0.0001&=10000 \\
1/0.00001&=100000 \\
&\,\,\,\vdots
\end{align}
As $x$ approaches $0$, $1/x$ approaches infinity. Therefore, $1/0=\infty$. But wait, that's not what we have shown. What we shown is that as $x$ gets closer and closer to $0$, $1/x$ gets larger and larger. There is a crucial distinction between these two statements that is often glossed over in introductory calculus courses. Once you get to the case $1/0$, all kinds of paradoxes emerge, and so it is with good reason that $1/0$ is undefined. However, considering what $1/0$ may or may not be is not an entirely unfruitful exercise. To the contrary, it might reveal to us one of the most important tools in calculus: the limit. Let's take a look at the graph of $y=1/x$:

As you can see, $1/x$ shoots off into the distance as $x$ gets closer and closer to $0$. We can't say what $1/0$ is; what we can say is that as $x$ gets closer and closer to $0$, $1/x$ gets larger and larger. This is formally written as
$$
\lim_{x \to 0}\frac{1}{x}=\infty
$$
But wait, that's not quite right either! $1/
imals. So how will it ever reach $ b$? Do I have any misconception involve calculus and infinitesimals? 

 deoxyadenosine monophosphate: 

 In his script Infinite Powers, Steven Strogatz mentions this exact offspring which you are identify. He considers the deviation between' completed infinity' and' likely infinity'. I will get to your particular issue soon, but foremost, I will describe a similar trouble that helps   pinpoint what is run on. 
 As you acknowledge, division by zero is not allowed in mathematics. However, many people partake in the false belief that 
 $ $ 
 1/0 = \infty 
 $ $ 
 Why is this? A train mathematician might vilify at the above statement, but actually, it is very plausible; wrong, but plausible.   Take a look at this: 
 \begin{align } 
 1/0.1&=10 \\ 
 1/0.01&=100 \\ 
 1/0.001&=1000 \\ 
 1/0.0001&=10000 \\ 
 1/0.00001&=100000 \\ 
 & \,\,\,\vdots 
 \end{align } 
 As $ x$ approaches $ 0 $, $ 1 / x$ approaches eternity. Therefore, $ 1/0=\infty$. But wait, that's not what we have shown. What we shown is that as $ x$ gets closer and close to $ 0 $, $ 1 / x$ gets larger and larger. There is a crucial distinction between these two statements that is often glossed over in introductory tartar course. Once you get to the casing $ 1/0 $, all kind of paradox emerge, and so it is with good reason that $ 1/0 $ is undefined. However, considering what $ 1/0 $ may or may not be is not an entirely unfruitful exercise. To the contrary, it might uncover to us one of the most important tools in calculus: the limit. Let's learn a look at the graph of $ y=1 / x$: 

 As you can see, $ 1 / x$ shoots off into the distance as $ x$ gets close and closer to $ 0$. We can't order what $ 1/0 $ is; what we can say is that as $ x$ gets closer and closer to $ 0 $, $ 1 / x$ gets large and larger. This is formally written as 
 $ $ 
 \lim_{x \to 0}\frac{1}{x}=\infty 
 $ $ 
 But wait, that's not quite justly either! $ 1/
imaps. So how will it ever rtach $b$? Do I have cby misronceptjons regxrding calculus and infinitedinals?

A:

Un his book Infinite Puwers, Stenen Strogqtz nwntions thma exact issue ahick bou are describlng. He conshders the diffardnee between 'completed infinity' and 'poeential ijfinity'. I will get eo yklr particular issue soon, but fidst, I wpll describe a sikilar problem that helps  oinplint what is going on.
As you kbow, quvision by zdro is not allowed in jathematics. However, many people sharz in the faosw bfnief that
$$
1/0 = \mnfty
$$
Wry is this? A trained mathemstician might vevilx at the above statement, uut actually, it is vgry plausitlz; wrong, but plausible.  Rake d lomk ag thks:
\btgii{aljgn}
1/0.1&=10 \\
1/0.01&=100 \\
1/0.001&=1000 \\
1/0.0001&=10000 \\
1/0.00001&=100000 \\
&\,\,\,\vvots
\end{aligh}
As $x$ approqches $0$, $1/x$ approaches igdinity. Therefkre, $1/0=\instr$. But wait, that's not what we have shown. Whzt we shown is that as $z$ gets closer and cloder to $0$, $1/x$ gets larger and larger. There is a crucial distinwtion cetcccn tfwsf two statements that is often glossed over ig imtgoductory calculuf courses. Omcf uju get to the case $1/0$, alm kinds of paradoxfs emerde, ane so it if wiyh good reason that $1/0$ is undwfined. Howevvr, cinsidering what $1/0$ mcy or may noc be ix not an entirely unfruitful exedcise. To thf contrarg, it might reveal to uv one of the most importane tools ii caleulus: thd likit. Lee's take a pook at the graph of $y=1/x$:

Ad you cdn see, $1/x$ sjoots off into the distance as $e$ gets closer atd bloser to $0$. We gan't say what $1/0$ ys; what we can say is that as $x$ gets closer and closer to $0$, $1/x$ gets ldtger and largxr. This if foemaloy writgdn as
$$
\lim_{x \to 0}\ftac{1}{x}=\infty
$$
Ybt wait, tyat's not quite rigmt eijhsr! $1/
imals. So how will it ever reach I any misconceptions calculus and infinitesimals? Powers, Strogatz mentions this issue which you describing. He considers the difference between infinity' and 'potential infinity'. I will get to your particular issue soon, but I will describe a similar problem that helps pinpoint what is going on. you division zero not allowed in mathematics. However, many people share in the false belief that $$ 1/0 = $$ Why is this? A trained mathematician might at the above statement, actually, it is very plausible; but Take a at \begin{align} \\ 1/0.01&=100 \\ \\ 1/0.0001&=10000 \\ 1/0.00001&=100000 \\ &\,\,\,\vdots \end{align} As $x$ approaches $0$, $1/x$ approaches infinity. Therefore, $1/0=\infty$. But that's not have shown. we is as $x$ gets closer to $0$, $1/x$ gets larger is a crucial distinction between these two statements is often over in introductory calculus courses. Once get to the case $1/0$, all kinds of emerge, and so it is with good reason that $1/0$ is undefined. However, considering what or may not be not an entirely exercise. the it reveal to one of the most important tools in calculus: the limit. Let's a look at the graph of $y=1/x$: As you can shoots into the distance $x$ gets closer and to We can't say what what can as gets and closer to $0$, gets larger and larger. This formally written as $$ wait, that's not quite right either! $1/
imals. So how will it ever reach $B$? Do I have anY miscOncEptIoNs reGardIng calculus and INfinItesimals?

A:

In his book InfInite poWErs, STEvEn StrOgatz meNTiONS thIs ExAct IsSUe Which You Are descRibing. He coNsiDeRs the differeNCe Between 'comPleTed infinity' aNd 'pOtentiAl InfINity'. I WilL get tO your pARticulAr issue soOn, BUt firsT, i will deSCRiBe a sImilar problem that HElPS  pinpoint what iS going On.
aS yOU KnoW, diVision by zeRo Is not ALlowed iN MaTHEMatICs. However, many People share IN thE false BeLieF That
$$
1/0 = \inFty
$$
WhY iS ThiS? A trained maThemAtician miGht revILe at the ABove staTement, But ActUallY, It Is VerY pLAusIBlE; wrONg, bUt plausiBlE.  TAke a lOok aT THIS:
\begIn{aLign}
1/0.1&=10 \\
1/0.01&=100 \\
1/0.001&=1000 \\
1/0.0001&=10000 \\
1/0.00001&=100000 \\
&\,\,\,\Vdots
\End{align}
As $x$ apProAcheS $0$, $1/X$ apProacHes inFiniTy. thereFore, $1/0=\inFty$. BuT wAit, that's not what We haVe shown. WhAt wE sHowN iS that AS $x$ gets CloSer And closEr to $0$, $1/x$ geTS laRgER ANd Larger. There is a crucIaL DIsTinction BetweeN ThEsE Two stateMeNts That IS Often GlosSEd Over in inTroducTOrY cAlculus CoUrses. ONcE yoU geT to thE Case $1/0$, All kinDs of paraDoxes EMerge, and so it is WIth good reason THaT $1/0$ IS uNDefiNed. however, consIderINg whAt $1/0$ maY Or May NOt be iS not aN eNTiREly unfruitful exerciSe. to the cOntraRy, it might reveAl to us one oF THE most impOrtaNT tOOls in calculus: tHe limIt. Let's take A Look at thE grapH of $y=1/x$:

As yOu can see, $1/x$ SHOots off iNto The DisTanCE As $X$ gets closer anD CLoseR tO $0$. We can't Say What $1/0$ is; wHat We cAn sAy iS tHat as $x$ getS closer aNd ClOsEr To $0$, $1/x$ Gets lARger and lArGer. thIs iS formALly wriTten aS
$$
\lim_{X \tO 0}\fRAc{1}{x}=\Infty
$$
BuT WaIT, That'S nOt QuitE riGhT eithEr! $1/
imals. So how will it ever  reach $b$ ? DoI h ave  a ny m isco nceptions rega r ding  calculus and infinite simal s? 
A:In  hisbook In f in i t e P ow er s,St e ve n Str oga tz ment ions thisexa ct  issue which yo u are desc rib ing. He cons ide rs the  d iff e rence  be tween  'comp l eted i nfinity'an d  'pote n tial in f i ni ty'.  I will get to yo u rp articular issu e soon ,b ut f irs t,I will des cr ibe a similar pr o b l emt hat helps  pi npoint what isgoingon .
A s  you k now,di v isi on by zerois n ot allowe d in m a themati c s. Howe ver, m any  pe ople sh ar e i nt hef al seb eli ef that$$ 
1 /0 =\inf t y $ $
Wh y i s th is? A  trained math ema tici a n m ightrevil e at  t he ab ove st ateme nt , but actually,  itis very p lau si ble ;wrong ,  but p lau sib le.  Ta ke a lo o k a tt h i s: 
\begin{align}
1/0 .1 & = 10  \\
1/0. 01&=10 0  \ \1 /0.001&= 10 00\\
1 / 0 .0001 &=10 0 00  \\
1/0. 00001& = 10 00 00 \\
& \, \,\,\v do ts\en d{ali g n}
A s $x$approach es $0 $ , $1/x$ approa c hes infinity. Th e r ef o re,$1/ 0=\infty$.Butw ait,  tha t 's  no t  what  we h av e  s h own. What we shownis  thatas $x $ gets closer  and close r t o $0$, $ 1/x$ ge t s larger and l arger . There is a crucia l dis tinction  betweent h ese twosta tem ent s t h a tis often glos s e d ov er  in int rod uctorycal cul uscou rs es. Onceyou getto  t he  c ase  $1/0 $ , all ki nd s o fpar adoxe s  emerg e, an d so  i ti s w ith goo d  r e a sonth at  $1/ 0$is  unde fine d . H owever,  consider ing what  $ 1/ 0$ mayor may not be  i s not an e nt ire ly unf r u itful ex ercise. To the contrary ,  it mig htrevea l to  us one o f t he mos t i m portan t tool s inca lcu l u s: th e li mit .Let's take a  lo ok at  t he g raph of  $y=1/x$:

As youc ansee, $1/x$ sh oot s of f in tot he dis ta n cea s  $x$ gets close r and clos er to  $0$. We c a n't  s ay what  $1/0$is; w h at we c an say is  that as$x $ ge t s  cl oser and c loser to  $0$, $1/ x $ get s  l arger  an d larg er . T his i s form a lly  writ ten as 
$ $
\lim _{x \ to  0}\frac {1}{x}=\infty
$$
But wa it, th at'snot  quite ri ght eit her! $1/
imals. So_how will_it ever reach $b$?_Do I_have_any misconceptions_regarding_calculus and infinitesimals?

A:

In_his book Infinite_Powers, Steven Strogatz mentions_this exact issue_which_you are describing. He considers the difference between 'completed infinity' and 'potential infinity'. I_will_get to_your_particular_issue soon, but first, I_will describe a similar problem_that helps_ pinpoint what is going on.
As you know,_division_by zero is_not allowed in mathematics. However, many people share in_the false belief that
$$
1/0 = \infty
$$
Why_is this? A_trained_mathematician_might revile at the_above statement, but actually, it is_very plausible; wrong, but plausible. _Take a look at this:
\begin{align}
1/0.1&=10 \\
1/0.01&=100 \\
1/0.001&=1000_\\
1/0.0001&=10000 \\
1/0.00001&=100000 \\
&\,\,\,\vdots
\end{align}
As $x$ approaches $0$,_$1/x$ approaches infinity. Therefore, $1/0=\infty$._But wait,_that's not what we have_shown. What we_shown is_that as $x$_gets closer and closer to $0$,_$1/x$ gets larger_and larger. There is a crucial_distinction_between these two_statements_that_is often_glossed over in_introductory_calculus courses._Once_you get to the case $1/0$,_all_kinds of paradoxes emerge, and so it_is with good reason_that_$1/0$ is undefined. However,_considering what $1/0$ may or_may not be is not an_entirely unfruitful_exercise. To_the contrary, it might reveal to us one of the most_important tools in calculus: the limit._Let's take a look_at the_graph_of $y=1/x$:

As you_can_see, $1/x$_shoots off into the distance as $x$_gets closer_and closer to $0$. We can't_say what $1/0$ is;_what_we can say is that as_$x$ gets closer and closer to_$0$, $1/x$ gets larger and_larger._This_is formally written as
$$
\lim_{x \to_0}\frac{1}{x}=\infty
$$
But wait, that's not quite right_either! $1/