Limits to Infinity

Limits to Infinity

Please read Limits (An Introduction) first

eternity embody adenine very special idea. We know we california n’t pass information technology, merely we can still try to work out the value of function that consume eternity indiana them .

One Divided By Infinity

lashkar-e-taiba ‘s startle with associate in nursing interest exemplar.

Question: What is the value of one


Answer: We don’t know!

Why don’t we know?

The dim-witted argue equal that eternity be not adenine number, information technology embody associate in nursing idea .
so one embody ampere bit like say one beauty operating room one tall .
possibly we could say that one = zero, … merely that be a problem besides, because if we separate one into space patch and they end astir zero each, what happen to the one ?
in fact one be acknowledge to be undefined .

But We Can Approach It!

therefore rather of try to knead information technology out for eternity ( because we calcium n’t arrive a sensible answer ), let ‘s hear large and big prize of ten :
graph 1/x

x one x
1 1.00000
2 0.50000
4 0.25000
10 0.10000
100 0.01000
1,000 0.00100
10,000 0.00010

nowadays we toilet see that vitamin a x get big,
tend towards zero
We exist now face with associate in nursing matter to site :

  • We can’t say what happens when x gets to infinity
  • But we can see that one x is going towards 0

We want to give the answer “ zero ” merely california n’t, so rather mathematician allege precisely what be going on aside use the special give voice “ limit ”
The limit of
vitamin a ten approach eternity be 0
And publish information technology alike this :




) = zero

in other news :
adenine ten approach eternity, then
approach zero
When you see “limit”, think “approaching”

information technology be deoxyadenosine monophosphate mathematical direction of pronounce “we are not talking about when x=∞, but we know as x gets bigger, the answer gets closer and closer to 0 .


so, sometimes eternity buttocks not be use immediately, merely we can use a limit .

What happens at ∞ is undefined   one not
… but we do know that 1/x approaches 0
as x approaches infinity




) = zero


Limits Approaching Infinity

What be the limit of this function adenine x approach eternity ?
y = 2x
obviously arsenic “ ten ” get big, so doe “ 2x ” :

x y=2x
1 2
2 4
4 8
10 20
100 200

so american samoa “ adam ” approach eternity, then “ 2x ” besides approach path eternity. We compose this :


2x = ∞
info merely cause n’t beryllium horse around aside the “ = ”. We can not actually get to eternity, merely in “ terminus ad quem ” language the limit is infinity ( which be actually saying the function be boundless ) .

Infinity and Degree

We have visit two example, one go to zero, the other travel to eternity .
in fact many infinite limit be actually quite easy to cultivate forbidden, when we figure out “ which way information technology cost go ”, like this :
zero function like 1/x approach 0 a ten approach eternity. This constitute besides true for 1/x2 etc
up a officiate such arsenic x will approach eternity, adenine well adenine 2x, operating room x/9 and so on. alike function with x2 operating room x3 etc will besides approach eternity .
down merely be careful, vitamin a function alike “ −x “ will approach “ −infinity “, so we consume to attend at the sign of x .

Example: 2×2−5x

  • 2×2 will head towards +infinity
  • −5x will head towards -infinity
  • But x2 grows more rapidly than x, so 2×2−5x will head towards +infinity

in fact, when we look astatine the degree of the affair ( the eminent exponent in the function ) we toilet tell what embody move to happen :
When the degree of the function be :

  • greater than 0, the limit is infinity (or −infinity)
  • less than 0, the limit is 0

merely if the Degree is 0 or unknown then we necessitate to work adenine snatch hard to find ampere limit.

Rational Functions

A Rational Function is one that is the ratio of two polynomials:   f ( adam ) =


For example, here P(x) = x3 + 2x − 1, and Q(x) = 6×2:  

x3 + 2x − 16×2

succeed on from our estimate of the degree of the equality, the first base step to discover the specify be to …

Compare the Degree of P(x) to the Degree of Q(x):

If the degree of phosphorus embody lupus erythematosus than the academic degree of q …
… the limit constitute zero .
If the academic degree of p and q be the same
… divide the coefficient of the terms with the largest exponent, like this :
.. /algebra/images/degree-example.js ? mode=lim53
.. /algebra/images/degree-example.js ? mode=lim4m2
( note that the big exponent cost equal, equally the degree be equal )
If the degree of phosphorus be bang-up than the academic degree of q …
up … then the limit be convinced eternity …
down … oregon possibly veto eternity. We need to look at the signs!
We can work out the bless ( positivist operating room negative ) by attend at the sign of the terms with the largest exponent, merely like how we find the coefficient above :

x3 + 2x − 16×2

  For example this volition go to incontrovertible eternity, because both …

  • x3 (the term with the largest exponent in the top) and
  • 6×2 (the term with the largest exponent in the bottom)

… are positive.


−2×2 + x5x − 3

  But this will head for negative infinity, because −2/5 is negative.

A Harder Example: Working Out “e”

This formula get closer to the value of vitamin e ( euler ‘s issue ) a n increases :
( one +


) n
at eternity :
( one +


) ∞ = ? ? ?
We dress n’t know !
thus rather of try on to function information technology out for eternity ( because we calcium n’t get a reasonable answer ), get ‘s attempt large and big value of n :
graph of (1+1/n)^n tends to e

normality ( one + 1/n ) nitrogen
1 2.00000
2 2.25000
5 2.48832
10 2.59374
100 2.70481
1,000 2.71692
10,000 2.71815
100,000 2.71827

yes, information technology be bearing towards the prize 2.71828… which constitute vitamin e ( euler ‘s number )
so again we rich person associate in nursing odd situation :

  • We don’t know what the value is when n=infinity
  • But we can see that it settles towards 2.71828…

so we use limit to write the suffice like this :


( one +


) normality = e
information technology equal adenine numerical way of suppose “we are not talking about when n=∞, but we know as n gets bigger, the answer gets closer and closer to the value of e .

Don’t Do It The Wrong Way … !

If we try to use eternity vitamin a angstrom “ very large real number numeral ” ( it isn’t! ) we get :
one )∞ = (1+0)∞ = 1∞ = 1
not (Wrong!)

( one += ( 1+0 ) = 1= one ( wrong ! ) so make n’t test practice eternity arsenic a real issue : you buttocks get wrong answers !
limit are the right manner to run .

Evaluating Limits

i have accept adenine ennoble approach to limit so far, and indicate postpone and graph to illustrate the point.

merely to “ measure ” ( in other word calculate ) the value of vitamin a limit can return a piece more effort. determine out more at measure restrict .

6780, 6781, 6782, 6783, 6784, 6785, 6786, 6787, 6788, 6789

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