Confusion over notation for finding limits

[feldspar]

I don't know how to properly present my answer to find the limit of a converging sequence like (1/2)^n.

I would just write something like this...

y=1/x+1, x=∞ } y=1/∞+1=0+1=1

but the syllabus gives something completely different and my textbooks don't seem to cover this portion of the syllabus.

This is the syllabus for A-level mathematics in the UK, presumably the equivalent of precalculus.
http://www.suffolkmaths.co.uk/pages/A-Level%20Activities/Menus/Pure.htm
press ctrl+f then type "finding" and you come across this

Iterative formulae

To include their use in finding of a limit L as n → ∞ by putting L = f(L)

I'm not entirely sure what it means by "by putting L = f(L)".

I get the impression this means substituting n for ∞ to find whether a sequence is periodic, divergent or convergent and it's limits if it has them but I have no idea what a "iterative formulae" is, neither do I know the proper notation I need to present my working. I can't find any examples of proper notation to follow and make sure I'm communicating my answer properly. I've been checking for quite a while now on the internet but I can't find any examples and I would appreciate some help.

 

[Mark44]

I don't know how to properly present my answer to find the limit of a converging sequence like (1/2)^n.

The sequence {(1/2)ⁿ}, n = 1, 2, ... converges, and in fact converges to zero.

I would just write something like this...

y=1/x+1, x=∞ } y=1/∞+1=0+1=1

This makes no sense. Is this supposed to be related to the sequence you started with? Either way, we NEVER substitute ∞ in an expression or equation. That's why we use limits.

but the syllabus gives something completely different and my textbooks don't seem to cover this portion of the syllabus.

This is the syllabus for A-level mathematics in the UK, presumably the equivalent of precalculus.
http://www.suffolkmaths.co.uk/pages/A-Level%20Activities/Menus/Pure.htm
press ctrl+f then type "finding" and you come across this

Iterative formulae

To include their use in finding of a limit L as n → ∞ by putting L = f(L)

I'm not entirely sure what it means by "by putting L = f(L)".

I get the impression this means substituting n for ∞ to find whether a sequence is periodic, divergent or convergent and it's limits if it has them but I have no idea what a "iterative formulae" is, neither do I know the proper notation I need to present my working. I can't find any examples of proper notation to follow and make sure I'm communicating my answer properly. I've been checking for quite a while now on the internet but I can't find any examples and I would appreciate some help.

 

[Ray Vickson]

I don't know how to properly present my answer to find the limit of a converging sequence like (1/2)^n.

I would just write something like this...

y=1/x+1, x=∞ } y=1/∞+1=0+1=1

but the syllabus gives something completely different and my textbooks don't seem to cover this portion of the syllabus.

This is the syllabus for A-level mathematics in the UK, presumably the equivalent of precalculus.
http://www.suffolkmaths.co.uk/pages/A-Level%20Activities/Menus/Pure.htm
press ctrl+f then type "finding" and you come across this

Iterative formulae

To include their use in finding of a limit L as n → ∞ by putting L = f(L)

I'm not entirely sure what it means by "by putting L = f(L)".

I get the impression this means substituting n for ∞ to find whether a sequence is periodic, divergent or convergent and it's limits if it has them but I have no idea what a "iterative formulae" is, neither do I know the proper notation I need to present my working. I can't find any examples of proper notation to follow and make sure I'm communicating my answer properly. I've been checking for quite a while now on the internet but I can't find any examples and I would appreciate some help.

It is not clear what is wanted because the notation is sloppy; I don't know if the person setting the problem was sloppy or whether you copied it out carelessly. The issue is: do you mean y = \frac{1}{x} + 1, \text{ or } y = \frac{1}{x+1}? If you don't use LaTeX, you must use brackets to enforce proper meaning; the first meaning would be written as y = (1/x) + 1 or y = 1 + 1/x, while the second one would be y = 1/(1+x). Depending on which you mean you will get different results when you take x → ∞.

RGV

 

[feldspar]

The sequence {(1/2)ⁿ}, n = 1, 2, ... converges, and in fact converges to zero.

do you mean y = \frac{1}{x} + 1, \text{ or } y = \frac{1}{x+1}?

I was trying to explain I wanted to know how to present a handwritten answer, I suppose I should just ask how to properly present the working and answers to these questions, the 2 questions are...

1: Find the limit of y = (\frac{1}{2})^x as x approaches infinity

2: Find the limit of y = \frac{1}{x} + 1 as x approaches infinity

 

[Mark44]

I was trying to explain I wanted to know how to present a handwritten answer, I suppose I should just ask how to properly present the working and answers to these questions, the 2 questions are...

1: Find the limit of y = (\frac{1}{2})^x as x approaches infinity

2: Find the limit of y = \frac{1}{x} + 1 as x approaches infinity

The work you show depends on how much detail your instructor wants to see. For example, if you need to use the definition of the limit in each problem, that's different from just evaluating each limit.

If all you need to do is evaluate each limit, both of these are pretty easy. The first has a limit of 0; the second has a limit of 1.

 

[feldspar]

I have to use proper notation to show I understand that as x approaches infinity, (1/x) reaches 0, therefore as x approaches infinity, (1/x) + 1 approaches 0 + 1.

If you were to divide up an apple infinite times you would end up with infinitely small shares which might as well be 0 unless you want to split hairs. Here is my attempt.

\lim_{x\to\infty}\frac{1}{x} + 1=(1/∞) + 1 = 0 + 1 = 1

The same could be said of the other equation, if you halve something infinite times you end up with an infinitely small share.

\lim_{x\to\infty}(\frac{1}{2})^x=(1^∞)/(2^∞) = (1/∞) = 0

I know infinity is a concept not a number but I don't know how else to express my ideas using mathematical notation.