x$ only approaches infinity* when $x$ approaches $0$ from the positive end. What if $x$ is a negative number that is getting closer and closer to $0$? Then, $1/x$ approaches $-$infinity. Perhaps we could write that
$$
\lim_{x \to 0}\frac{1}{x}=\pm\infty
$$
but mathematicians like it if limits have a single, definite value. Therefore, both of the above statements are incorrect, and what we should be writing is this:
$$
\lim_{x \to 0^+}\frac{1}{x}=\infty \text{ and } \lim_{x \to 0^-}\frac{1}{x}=-\infty
$$
(The little '$+$' right next to the $0$ indicates that we are considering the case where $x$ is a positive number that is approaching $0$. Likewise, the '$-$' means $x$ is a negative number that is approaching $0$.)
Don't worry if all of the little details don't make complete sense to you. Just try to remember these two key facts:

In calculus, mathematicians work with limits.
When trying to compute limits such as $1/x$ as $x$ approaches $0$, the fact that $1/0$ is undefined is neither here nor there. We are not trying to work out what happens when we 'get to zero'. Rather, we are looking at what happens as we move closer and closer to $0$, both from the positive and negative direction.**

Now let's compare what we have learnt about limits with what we think we know about infinitesimals. The biggest problem with the concept of an infinitesimal in my mind is that they suggest that there is a'smallest possible number'. Actually, when we are working with the standard real numbers, there is no such thing. This should be intuitively obvious: however low you go, you can always go lower. You might also be sympathetic to the idea that
$$
1/\text{infinitesimal}=\infty
$$
Again, this is problematic, not least because it treats infinity as if it were a number. Therefore, we should be extremely cautious when someone mentions the words 'infinitesimal', or 'infinitely small'. Often when they do, they are using these terms as a mere shorthand for the limits we worked with earlier. For instance, if I write 'as $x
x$ only approaches infinity * when $ x$ approaches $ 0 $ from the incontrovertible goal. What if $ x$ is a negative number that is getting close and closer to $ 0 $? Then, $ 1 / x$ approaches $ -$infinity. possibly we could write that 
 $ $ 
 \lim_{x \to 0}\frac{1}{x}=\pm\infty 
 $ $ 
 but mathematicians like it if limits suffer a single, definite value. Therefore, both of the above statement are incorrect, and what we should be writing is this: 
 $ $ 
 \lim_{x \to 0^+}\frac{1}{x}=\infty \text { and } \lim_{x \to 0 ^ -}\frac{1}{x}=-\infty 
 $ $ 
 (The fiddling' $ + $' justly next to the $ 0 $ indicates that we are considering the case where $ x$ is a positive number that is approach $ 0$. Likewise, the' $ -$' means $ x$ is a negative issue that is approaching $ 0$.) 
 Don't worry if all of the little details don't stool complete sense to you. Just try to remember these two cardinal facts: 

 In calculus, mathematicians work with limits. 
 When trying to calculate limit such as $ 1 / x$ as $ x$ approaches $ 0 $, the fact that $ 1/0 $ is undefined is neither here nor there. We are not trying to work out what happens when we' get to zero'. Rather, we are looking at what happens as we move nearer and close to $ 0 $, both from the positive and minus steering. * * 

 Now let's compare what we have learnt about terminus ad quem with what we remember we know about infinitesimals. The boastful problem with the concept of an infinitesimal in my mind is that they suggest that there is a'smallest possible number'. Actually, when we are work with the standard real numbers, there is no such thing. This should be intuitively obvious: however humble you start, you can always go lower. You might also be sympathetic to the idea that 
 $ $ 
 1/\text{infinitesimal}=\infty 
 $ $ 
 Again, this is debatable, not least because it treats infinity as if it were a number. consequently, we should be extremely cautious when someone note the words' infinitesimal', or' infinitely little'. Often when they act, they are using these terms as a bare shorthand for the limits we worked with earlier. For instance, if I publish' as $ x
x$ ojly approaches infinity* dhen $x$ approachgs $0$ from the pksitive dnd. What if $x$ is a negative iumbwr thqt is getting closer avd closer to $0$? Theb, $1/x$ epproaches $-$infinmfy. Perhaps we gould xrite that
$$
\lim_{x \jo 0}\frac{1}{x}=\pm\inxty
$$
but mathemadizicns like it if limits have a single, qefinitr galue. Thereforg, botn of fhe above statements are incorrect, and whet we should be writing is this:
$$
\lim_{x \to 0^+}\frwc{1}{x}=\ijfty \text{ and } \lim_{d \to 0^-}\frac{1}{x}=-\ibfty
$$
(Eye little '$+$' rkght next uo the $0$ indidates that we are considering tfe caxe where $x$ iw a kositive numuer thwt is approagning $0$. Nikewisr, the '$-$' means $x$ is a begative number that ms approaching $0$.)
Don't rorry if dlm of the little dwtqils gon't makd coopltte ssnse tl yku. Just trg to remembwr these two key favtf:

Pm calculus, mzthemaeisians work with limits.
When trying to cokpufe limits such as $1/x$ as $z$ approaches $0$, the facj that $1/0$ is undefined is neither here nor there. We are not tsying go cirk ojr ahat happens when we 'get to zero'. Rather, we arq lpoling at what hcppens as we movr flpfer and closet to $0$, both from the positive and nedativw directijn.**

Noe let's compare what we have learnt aboun linits with what we chink we knoc abouj infimitesimals. The biggest 'roblej with the foncept or an infinitesimau ik mf mind is that they suggese that thxre ix a'smaluest possifle number'. Actually, when we are wogking whth the stwndard real numbers, there is no such thing. Thos shmuld be nntuitlvely obvious: hjwever low you go, you can xlways go mower. Ymu might alfo be sympathajic to the idxa that
$$
1/\tevt{indiniresimal}=\kvfty
$$
Again, this is problvmctic, not oeast because it tveats jnfinity as if nu wtrw a number. Thetefure, wv siould te extremely caugiojx whev someone mcntkons the words 'infiniteshmal', or 'infinitely smakl'. Often wyen they do, they are ising these terms ws a kerx shoryhagd for the limits we worked wifh earlieg. Fjr instance, yf I write 'as $x
x$ only approaches infinity* when $x$ approaches the end. What $x$ is a closer closer to $0$? $1/x$ approaches $-$infinity. we could write that $$ \lim_{x 0}\frac{1}{x}=\pm\infty $$ but mathematicians like it if limits have a single, definite value. both of the above statements are incorrect, and what we should be writing this: \lim_{x 0^+}\frac{1}{x}=\infty and } \lim_{x \to 0^-}\frac{1}{x}=-\infty $$ (The little '$+$' right next to the $0$ indicates that are considering the case where $x$ is a number that is approaching Likewise, the '$-$' means $x$ a number that approaching Don't if all of little details don't make complete sense to you. Just try to remember these two key facts: In mathematicians work When trying compute such $1/x$ as $x$ the fact that $1/0$ is undefined nor there. We are not trying to work what happens we 'get to zero'. Rather, we looking at what happens as we move closer closer to $0$, both from the positive and negative direction.** Now let's compare what we about limits with what think we know infinitesimals. biggest with concept of infinitesimal in my mind is that they suggest that there is possible number'. Actually, when we are working with the standard there no such thing. should be intuitively obvious: low go, you can always You also the that 1/\text{infinitesimal}=\infty $$ Again, this problematic, not least because it infinity as if it should be extremely cautious when someone mentions the 'infinitesimal', or 'infinitely small'. Often when they they are using these terms as a mere shorthand for the limits worked with instance, if I write 'as $x
x$ only approaches infinity* whEn $x$ approacHes $0$ frOm tHe pOsItivE end. what if $x$ is a negaTIve nUmber that is getting closEr and ClOSer tO $0$? thEn, $1/x$ apProacheS $-$InFINitY. PErHapS wE CoUld wrIte That
$$
\lim_{X \to 0}\frac{1}{x}=\pm\InfTy
$$
But mathematiCIaNs like it if LimIts have a singLe, dEfinitE vAluE. thereForE, both Of the aBOve staTements arE iNCorrecT, And what WE ShOuld Be writing is this:
$$
\liM_{X \tO 0^+}\Frac{1}{x}=\infty \text{ And } \lim_{X \tO 0^-}\FrAC{1}{X}=-\inFty
$$
(the little '$+$' rIgHt nexT To the $0$ inDIcATES thAT we are consideRing the case WHerE $x$ is a pOsItiVE numbeR that Is APprOaching $0$. LikeWise, The '$-$' means $x$ Is a negATive numBEr that iS approAchIng $0$.)
don't WOrRy If aLl OF thE LiTtlE DetAils don't MaKe ComplEte sENSE To yoU. JuSt trY to reMember these twO keY facTS:

In CalcuLus, maThemAtIcianS work wIth liMiTs.
When trying to cOmpuTe limits sUch As $1/X$ as $X$ aPproaCHes $0$, the FacT thAt $1/0$ is undEfined iS NeiThER HErE nor there. We are not tRyING tO work out What haPPeNs WHen we 'get To ZerO'. RatHER, we arE looKInG at what hAppens AS wE mOve closEr And cloSeR to $0$, BotH from THe poSitive And negatIve diREction.**

Now let's cOMpare what we haVE lEARnT AbouT liMits with whaT we tHInk wE knoW AbOut INfiniTesimAlS. thE Biggest problem with tHe ConcepT of an Infinitesimal In my mind is THAT they sugGest THaT There is a'smalleSt posSible numbeR'. actually, When wE are workIng with thE STandard rEal NumBerS, thERE iS no such thing. THIS shoUlD be intuItiVely obvIouS: hoWevEr lOw You go, you cAn always Go LoWeR. YOu mIght aLSo be sympAtHetIc To tHe ideA That
$$
1/\teXt{infInitEsImAL}=\inFty
$$
AgaiN, ThIS Is prObLeMatiC, noT lEast bEcauSE it Treats iNfinity as If iT Were A nUmBer. TherEfore, we should Be Extremely cAuTioUs when SOMeone menTions the words 'infinitesiMAl', or 'infIniTely sMall'. often when TheY do, theY arE Using tHese teRms as A mEre SHOrthaND FoR thE lImits we worKED wiTh earLiEr. FoR instanCe, if I write 'as $x
x$ only approaches infinit y* when $x $ app roa che s$0$from  the positivee nd.What if $x$ is a negat ive n um b er t h at  is g ettingc lo s e r a nd  c los er to  $0$?  Th en, $1/ x$ approac hes  $ -$infinity.P er haps we co uld  write that$$\lim_{ x\to 0}\fr ac{ 1}{x} =\pm\i n fty
$$ 
but math em a tician s  like i t if  lim its have a single ,  d e finite value.Theref or e ,b o thofthe abovest ateme n ts arei nc o r r ect ,  and what weshould be w r iti ng isth is: $$
\li m_{x\t o  0^ +}\frac{1}{ x}=\ infty \te xt{ an d  } \lim _ {x \to0^-}\f rac {1} {x}= - \i nf ty$$ (Th e  l itt l e ' $+$' rig ht  n ext t o th e $ 0 $ in dic ates  that  we are consi der ingt hecasewhere  $x$  i s a p ositiv e num be r that is appro achi ng $0$. L ike wi se,  t he '$ - $' mea ns$x$  is a n egative num be r t ha t is approaching $ 0$ . ) 
D on't wor ry ifa ll  o f  the lit tl e d etai l s  don' t ma k ecomplete  sense to  y ou. Jus ttry to  r eme mbe r the s e tw o keyfacts:
In ca l culus, mathema t icians work w i th l im i ts.Whe n trying to  com p utelimi t ssuc h  as $ 1/x$as $x $  approaches $0$, th efact t hat $ 1/0$ is undef ined is ne i t h er herenort he r e. We are nottryin g to worko ut whathappe ns whenwe 'get t o zero'. R ath er,  we  ar e lo oking at what h appe ns  as wemov e close r a ndclo ser  t o $0$, bo th fromth epo si tiv e and negative  d ire ct ion .**
N ow let 's co mpar ewh a t w e havel ea r n t ab ou tlimi tswi th wh at w e  th ink weknow abou t i n fini te si mals. T he biggest pr ob lem with t he  co ncepto f  an infi nitesimal in my mind is that th eysugge st t hat there  is  a'sma lle s t poss ible n umber '.  Ac t u ally, w he n w eare workin g wit h the  s tand ard rea l numbers, there i s  no  such thing.Thi s sh o u ld  be in t uit iv e lyo b vious: howeverlow you go ,y ou  can alway s  go  l ower. Y ou migh t als o  be sym patheticto the id ea  tha t $$1/\text{in finitesi mal}=\inf t y
$$A ga in, t his  is pr ob lem atic,  not l e ast  beca use it  t reatsinfin it y as ifit were a number. There fore,we sh oul d be extr eme l y c autious w hensomeone me nti ons  thewor d s 'in fini t es ima l ', or  'in f initely s m al l'. O ft en when the y d o,theyare usingthes e terms as a mere shorthand forthel i mit s w e  wor ke d with earlier . F or i nstance,  i f I write ' as $x
x$ only_approaches infinity*_when $x$ approaches $0$_from the_positive_end. What_if_$x$ is a_negative number that_is getting closer and_closer to $0$?_Then,_$1/x$ approaches $-$infinity. Perhaps we could write that
$$
\lim_{x \to 0}\frac{1}{x}=\pm\infty
$$
but mathematicians like it if_limits_have a_single,_definite_value. Therefore, both of the_above statements are incorrect, and_what we_should be writing is this:
$$
\lim_{x \to 0^+}\frac{1}{x}=\infty \text{_and_} \lim_{x \to_0^-}\frac{1}{x}=-\infty
$$
(The little '$+$' right next to the $0$ indicates_that we are considering the case_where $x$ is_a_positive_number that is approaching_$0$. Likewise, the '$-$' means $x$_is a negative number that is_approaching $0$.)
Don't worry if all of the_little details don't make complete sense_to you. Just try to_remember these_two key facts:

In calculus, mathematicians_work with limits.
When_trying to_compute limits such_as $1/x$ as $x$ approaches $0$,_the fact that_$1/0$ is undefined is neither here_nor_there. We are_not_trying_to work_out what happens_when_we 'get_to_zero'. Rather, we are looking at_what_happens as we move closer and closer_to $0$, both from_the_positive and negative direction.**

Now_let's compare what we have_learnt about limits with what we_think we_know about_infinitesimals. The biggest problem with the concept of an infinitesimal in_my mind is that they suggest_that there is a'smallest_possible number'._Actually,_when we are_working_with the_standard real numbers, there is no such_thing. This_should be intuitively obvious: however low_you go, you can_always_go lower. You might also be_sympathetic to the idea that
$$
1/\text{infinitesimal}=\infty
$$
Again, this_is problematic, not least because_it_treats_infinity as if it were_a number. Therefore, we should be_extremely cautious when_someone mentions the words 'infinitesimal', or 'infinitely_small'._Often when they do, they are_using_these terms as a mere shorthand_for_the_limits we worked with earlier._For instance, if I write 'as_$x