Naturals or Reals When Taking Limits to Obtain the Value of Euler's Number e?

In summary, the existence of a limit for a function at a point can be determined by checking if all sequences that approach that point also have a limit at that point. This criterion can also be used to prove the non-existence of a limit by finding two different sequences that approach the point and have different limits. This is helpful in determining the continuity of a function, as it can be shown by using sequences that the limit at a point does not exist.
  • #1
mcastillo356
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TL;DR Summary
I've found two ways of taking limits in order to obtain ##e## number. Do they belong to naturals in one case and to the reals in the second case?
Hi PF

Searching on the Internet, I've found this definition:

Definition: Euler's Number as a Limit

(i) ##e=\displaystyle\lim_{x\to{0}}{(1+x)^{\displaystyle\frac{1}{x}}}##

and

(ii) ##e=\displaystyle\lim_{n\to{\infty}}{(1+\displaystyle\frac{1}{n})^n}##

Questions:
1-Does it make sense Definition (i)? I don't think so: ##(1+x)## on the base is strange: a lonely ##x## on the sum.
2-If both make sense, does ##x\in{\mathbb{R}}## in (i), and ##n\in{\mathbb{N}}## in (ii)?.

Attempt: I think I've fallen into an erratic web; but let's suppose the contrary. In that case, no matters reals neither naturals. I make no distinction. It could be also ##x\in{\mathbb{N}}##, and ##n\in{\mathbb{R}}##.
 
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  • #2
We have the following general proposition:

If [itex]\lim_{x \to c} f(x)[/itex] exists and [itex](x_n)[/itex] is a real sequence such that [itex]\lim_{n \to \infty} x_n = c[/itex] then [tex]
\lim_{n \to \infty} f(x_n) = \lim_{x \to c} f(x).[/tex] This holds also for one-sided limits, provided [itex]x_n \to c[/itex] from the appropriate direction.
 
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  • #3
mcastillo356 said:
Definition: Euler's Number as a Limit
(i) ##e=\displaystyle\lim_{x\to{0}}{(1+x)^{\displaystyle\frac{1}{x}}}##
and
(ii) ##e=\displaystyle\lim_{n\to{\infty}}{(1+\displaystyle\frac{1}{n})^n}##
mcastillo356 said:
1-Does it make sense Definition (i)? I don't think so: ##(1+x)## on the base is strange: a lonely ##x## on the sum.
2-If both make sense, does ##x\in{\mathbb{R}}## in (i), and ##n\in{\mathbb{N}}## in (ii)?.
1. Both are equally valid definitions of the number ##e##. Any positive real number not equal to 1 can be a base of an exponential expression. A negative base results in problems with continuity.

I don't understand what "a lonely ##x## on the sum" means.

2. Yes. For the first definition, x is a real number "close to" 1. In the second, n is an integer.
 
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  • #4
pasmith said:
We have the following general proposition:

If [itex]\lim_{x \to c} f(x)[/itex] exists and [itex](x_n)[/itex] is a real sequence such that [itex]\lim_{n \to \infty} x_n = c[/itex] then [tex]
\lim_{n \to \infty} f(x_n) = \lim_{x \to c} f(x).[/tex] This holds also for one-sided limits, provided [itex]x_n \to c[/itex] from the appropriate direction.
In fact, an alternative, equivalent definition of a limit uses sequences:

The function ##f## has limit ##L## at the point ##a## iff for all sequences ##\{x_n\} \in D - \{a\}## (where ##D## is the domain of ##f##) with ##\lim_{n \to \infty} x_n = a## we have ##\lim_{n \to \infty} f(x_n) = L##.

This can be shown to be equivalent to the more usual epsilon-delta definition. I think it's enormously useful, especially if you want to prove that a limit does not exist.
 
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  • #5
Mark44 said:
1. Both are equally valid definitions of the number ##e##.
Perfect.
Mark44 said:
Any positive real number not equal to 1 can be a base of an exponential expression. A negative base results in problems with continuity.

I don't understand what "a lonely ##x## on the sum" means.

2. Yes. For the first definition, x is a real number "close to" 1. In the second, n is an integer.
What means the first sentence? ##1^5## is not an exponential expression? Why can't ##1## be a base? Is it because is iterative? I mean, no matter the exponent, always equals ##1##.
"a lonely ##x## on the sum" is an unfortunate thought. I refer to the first quote (#1)
In the second, n is an integer is mentioned. Sequences aren't meant to deal with naturals?

Working on #2 and #4 post. Not sure to achieve:cry:
 
  • #6
mcastillo356 said:
What means the first sentence? ##1^5## is not an exponential expression? Why can't ##1## be a base? Is it because is iterative? I mean, no matter the exponent, always equals ##1##.
Yes, ##1^5## is an exponential expression, but as you note, ##1^5## is just a convoluted way to write 1. What I meant was that when we talk about the base of an exponential expression, we usually exclude 1 as a possible base because ##1^x## isn't very interesting. All of the other exponential functions, where the base is positive but not equal to 1, have inverses -- ##\log## functions.
mcastillo356 said:
Sequences aren't meant to deal with naturals?
Yes, a sequence can be defined as a function whose domain is the natural numbers, but the 2nd definition you showed is a limit, not a sequence. You could turn it into a sequence by the substitution ##n = \frac 1 x##, where x is chosen so that its reciprocal is an integer.
 
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  • #7
Mark44 said:
Yes, ##1^5## is an exponential expression, but as you note, ##1^5## is just a convoluted way to write 1. What I meant was that when we talk about the base of an exponential expression, we usually exclude 1 as a possible base because ##1^x## isn't very interesting. All of the other exponential functions, where the base is positive but not equal to 1, have inverses -- ##\log## functions.
Absolutely sound paragraph. Number ##1## is tricky, slippery, and a ground for brainy mathematicians.

Mark44 said:
Yes, a sequence can be defined as a function whose domain is the natural numbers, but the 2nd definition you showed is a limit, not a sequence. You could turn it into a sequence by the substitution ##n = \frac 1 x##, where x is chosen so that its reciprocal is an integer.
Brilliant! Good argument, helpful to judge, decide about, and eventually deal with number ##1##
 
  • #8
Hi, PF, I'm working on second and fourth posts of the thread. Second is very interesting, and #4 seems to be an alternative proposal; that is, they state the same argument.
Next I write my point of view, after a quickly searching. Actually, is a copy and paste. Finally, I'll ask the doubt. Here it goes:
Limit and sequences
We consider the set of all sequences ##\{x_n\}## such that

a) ##x_n\neq{a}\;\forall{n}##
b) ##\lim_{n\to\infty}{x_n=a}##

The fofllowing characterization of existence of limit verifies:
A function ##f## has got limit ##l## at a point ##a## iff for any sequence ##x_n## satisfying a) and b) it verifies ##\lim_{n\to\infty}{f(x_n)=l}##
The previous condition is useful to prove that some functions do not have limit at a point ##a##; it is enough to find a sequence ##x_n## which elements are different to ##a##, such that ##\lim_{n\to\infty}{x_n=a}##, and ##\lim_{n\to\infty}{f(x_n)}## doesn't exist. Other way to use the previous criterion in the opposite sense is to find two different sequences ##x_n## and ##y_n##, verifying a) and b) properties, such that
##\lim_{n\to\infty}{f(x_n)}\neq{\lim_{n\to\infty}{f(y_n)}}##

Example
Consider ##f(x)=\sin{(1/x)}## for ##x\neq{0}##. Prove it has no limit at ##x=0##, making use of sequences:

Take ##x_n## and ##y_n##, convergent to ##0##, such that
##x_n=\displaystyle\frac{1}{2\pi n}##
and
##y_n=\displaystyle\frac{1}{\dfrac{\pi}{2}+2\pi n}##, ##n\in{\mathbb{N}}##
As ##f(x_n)=\sin{\Big(\dfrac{1}{x_n}\Big)}=\sin{(2\pi n)}=0\;\forall{n}##, we have that ##\lim_{n\to\infty}{f(x_n)}=0##
On the other hand, ##f(y_n)=\sin{\Big(\dfrac{1}{y_n}\Big)}=\sin{\Big(\dfrac{\pi}{2}+2\pi n\Big)}=1\;\forall{n}##; therefore ##\lim_{n\to\infty}{f(y_n)}=1##.

This proves that function ##f##, whose graph is shown, has got no limit at ##0##.

Doubt
Is this speech near to stated at 2nd and fourth posts? IMG_20221201_213321.jpg

Sorry, bad LaTeX. Hope is understandable
 
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  • #9
Re
mcastillo356 said:
TL;DR Summary: I've found two ways of taking limits in order to obtain ##e## number. Do they belong to naturals in one case and to the reals in the second case?

Hi PF

Searching on the Internet, I've found this definition:

Definition: Euler's Number as a Limit

(i) ##e=\displaystyle\lim_{x\to{0}}{(1+x)^{\displaystyle\frac{1}{x}}}##

and

(ii) ##e=\displaystyle\lim_{n\to{\infty}}{(1+\displaystyle\frac{1}{n})^n}##

Questions:
1-Does it make sense Definition (i)? I don't think so: ##(1+x)## on the base is strange: a lonely ##x## on the sum.
2-If both make sense, does ##x\in{\mathbb{R}}## in (i), and ##n\in{\mathbb{N}}## in (ii)?.

Attempt: I think I've fallen into an erratic web; but let's suppose the contrary. In that case, no matters reals neither naturals. I make no distinction. It could be also ##x\in{\mathbb{N}}##, and ##n\in{\mathbb{R}}##.
Remember that naturals are Real numbers . Idea in both can be seen as compounding a very small number infinitely -many times. So one , the 1+1/x will be "pulling " to remain close to 0, while the exponent will "pull" in the opposite direction.
To verify these are equivalent, try the change of variable 1/n =x .

Is that what you were asking?
 
  • #10
WWGD said:
Re

Remember that naturals are Real numbers . Idea in both can be seen as compounding a very small number infinitely -many times. So one , the 1+1/x will be "pulling " to remain close to 0, while the exponent will "pull" in the opposite direction.
To verify these are equivalent, try the change of variable 1/n =x .

Is that what you were asking?
Very enriching words. But I was asking the Forum to check my point of view about these two limits to obtain ##e##, which I consider as a ##\lim_{x\rightarrow{\infty}}##, for ##x\in{\mathbb{R}}##, on one side, and the equivalent way to reach ##e##, through the limit of a sequence, ##\Big(1+\frac{1}{n}\Big)^n##, on the other side; this second approach is made within ##n\in\mathbb N##?.
 
  • #11
mcastillo356 said:
Very enriching words. But I was asking the Forum to check my point of view about these two limits to obtain ##e##, which I consider as a ##\lim_{x\rightarrow{\infty}}##, for ##x\in{\mathbb{R}}##, on one side, and the equivalent way to reach ##e##, through the limit of a sequence, ##\Big(1+\frac{1}{n}\Big)^n##, on the other side; this second approach is made within ##n\in\mathbb N##?.

The limit [tex]
\lim_{x \to \infty} (1 + x^{-1})^x[/tex] can also be taken through real values of [itex]x[/itex], and a proposition cited earlier then leads to [itex]\lim_{n \to \infty} (1 + x_n^{-1})^{x_n} = e[/itex] for any sequence [itex]x_n \to \infty[/itex], and one is free to choose [itex]x_n = n[/itex].
 
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  • #12
In general, if a countable sequence of function values converge and has a limit, it does not imply that the function itself has a limit. Example: [itex]\lim_{n\rightarrow \infty}( \sin(n\cdot\pi)) = 0 [/itex] since all elements in the sequence are 0, but [itex]\lim_{x\rightarrow \infty}( \sin(x)) [/itex] does not exist.
 
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1. What is Euler's number e?

Euler's number e is a mathematical constant that is approximately equal to 2.71828. It is an important number in calculus and is often used in exponential and logarithmic functions.

2. How is Euler's number e related to natural and real numbers?

Euler's number e is a real number, meaning it can be expressed as a decimal. It is also a natural number, as it is the base of the natural logarithm function. This function is used to find the value of e when taking limits.

3. What is the significance of taking limits to obtain the value of e?

Taking limits is a common mathematical technique used to find the value of a function as it approaches a certain point. In the case of Euler's number e, taking limits allows us to find the exact value of e, which cannot be expressed as a finite decimal.

4. How is Euler's number e calculated using limits?

The value of e can be calculated by taking the limit of (1 + 1/n)^n as n approaches infinity. This limit is equal to e, and the larger the value of n, the more accurate the calculation will be.

5. What are some real-world applications of Euler's number e?

Euler's number e has many applications in mathematics, physics, and engineering. It is used to model growth and decay in natural systems, as well as in financial calculations such as compound interest. It also appears in the normal distribution, which is used to describe many real-world phenomena.

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