# Avoiding the limit notation during long algebraic manipulations

For example, look at this:

\begin{align} \lim_{x \rightarrow \infty}\frac{x}{x+1} &= \lim_{x \rightarrow \infty}{\frac{\frac{x}{x}}{\frac{x}{x}+\frac{1}{x}}}\\ &= \lim_{x \rightarrow \infty}{\frac{1}{1+0}} \\ &= \lim_{x \rightarrow \infty}{\frac{1}{1}} \\ &= \lim_{x \rightarrow \infty}{1} \\ &= 1 \\ \end{align}

Maybe this isn't the best example as most of those steps are unnecessary, but sometimes you end up with something like that, having to write $\lim_{x \rightarrow \infty}$ over and over again. Is there any notation or method one can use to avoid having to do this?

• You can first manipulate the expression using only 'proper' equalities (equailities that don't involve limits) and only then take the limit. You can write $\lim$ without the $x\rightarrow \infty$ (but then you have to remember that $x$ tends to $\infty$). Also, Once you reach something of the form $\lim_{x\rightarrow \infty} \frac{1}{1+0}$ then it is widely accepted to just write it equates to $1$ (Without further explaining). I assume this comment wasn't helpful. – Ranc Oct 17 '16 at 17:10
• Like @Ranc says, you could just do the formal manipulations, then come back and do the limits. It is visually cleaner and may be easier for you to understand. If I was grading, I wouldn't take any points off for this. Hopefully no one else would either. – Cameron Williams Oct 17 '16 at 17:12
• If this is for a course, a tidy but relatively safe route would be to write in words something like "all of these lines should begin with $\displaystyle{\lim_{x\to\infty}}$". – Mark S. Oct 17 '16 at 22:57
• A rule of thumb I always use is: make sure the equalities you write are true without writing the limit during the algebraic manipulation, and remove the symbol of limit when actually calculating it (and when it is clear to remove it) in your example I would have written $\displaystyle{\lim_{x\to\infty}\dfrac{x}{x+1}=\lim_{x\to\infty}\dfrac{\frac{x}{x}}{\frac{x}{x}+\frac{1}{x}}=\lim_{x\to\infty}\dfrac{1}{1+\frac{1}{x}}=\dfrac{1}{1+0}=1.}$ – Darío G Oct 17 '16 at 23:03
• Remember that the goal of writing math (indeed, writing anything) is to communicate as clearly as possible. The goal is not to write the fewest number of characters. For this reason, I discourage you from using any such shortcut; it will detract from clarity. Also, @Wore's comment is an excellent one, in that it's mathematically correct as well as clear: in the OP, $\frac1x$ was replaced by $0$ inside the limit, which is an unjustified mathematical step. – Greg Martin Oct 18 '16 at 1:39

Having commented, edited and almost commented again, I realize it is better if I will write everything in an organized manner.

1. Manipulate the expression using only 'proper' equalities (equailities that don't involve limits) and only then take the limit. For example: $$\frac{x+1}{x} = 1 + \frac{1}{x} \xrightarrow[x\rightarrow\infty]{} 1$$
2. If we were to look at your example, we could essentially rewrite it similarly to what we did in point 1, without writing $\text{lim}$ so many times: $$\frac{x}{x+1}=\frac{\frac{x}{x}}{\frac{x}{x}+\frac{1}{x}}\xrightarrow[x\rightarrow\infty]{}\frac{1}{1+0}={\frac{1}{1}}=1$$
3. Write $\text{lim}$ without the subscript '$x→∞$' (but then you have to remember that $x$ tends to $∞$).
4. Once you reach something of the form $\lim_{x\rightarrow \infty}\frac{1}{1+0}$ then it is widely accepted to just write it equates to $1$ (Without further explaining).
5. Sometimes it is of great convinience to use O-notation. Use of that little-O would look like: $$\frac{x+1}{x} = 1+ o(1)\quad(x\rightarrow\infty)$$ And big-O: $$\frac{x+1}{x} =1+\mathcal{O}\left(\frac{1}{x}\right)\quad(x\rightarrow\infty)$$ Having brought the expression to a final form (which resembles the 2 latter equalities) one immediately recognizes the limit. This notation, usually, conceals some limiting procedure.

Of course, the above guidelines only suggest a way of having a cleaner, more compact paper, and are not of any mathematical importance. The basic rule is; the purpose of mathematical notations is to convey an ideas or objects. [I would say] As a rule of thumb: Notations are good if they are intuitive (and intuitevely understandable), neat and simple.

• Thanks for the great answer! A few questions: for your point 1, wouldn't I write $x \rightarrow \infty$ below the arrow, not above it? For point 2, is it universally recognized to be valid to leave the subscript out? (I've never seen it left out in a book before). For 3, isn't it called Big-O notation? If it is, you motivated me to look it up :) seems quite useful, and I see it used in a lot of different places (forgive my naïveté). – Skeleton Bow Oct 17 '16 at 17:40
• @SkeletonBow, 1) As long as it understood, it usually doesn't quite matter, but I think you're right and it is more common as subscript (In my real analysis notes and functional analysis notes it is written with subscript :) ). 2) Again, if it is understood, and there is no virtual reason for the limit to change (by a change of variables) then leave it out and have a cleaner paper. If it is a limit where you deal with many variables or change the limit frequently - keep the subscript. 3) Theres Big-O and Little-O notations, I've used them both. – Ranc Oct 17 '16 at 17:44
• @SkeletonBow, Why do we write $x \rightarrow \infty$? We indicate at least 2 things: 1) $x$ is the variable involved in the limiting process. 2) We are interested in what's going on when we're tending to $\infty$. If it is clear that it is $x$ we're working with (and it is, since it is the only variable) and it is clear that $x$ is tending to $\infty$ (and it is, since we have reasonable short-term memory): then why not just write $\lim$? – Ranc Oct 17 '16 at 17:47
• @SkeletonBow, lol, sure thing. I'm glad my ever-expanding comment grew up into something useful. – Ranc Oct 17 '16 at 18:05
• The Big-Oh notation is tricky, in particular in 5. you write $o(1) = O(1/x)$ which is wrong. Although that term in fact is $1/x$, once you are down to $o(1)$ you don't know if it is some other term with different convergence speed. You could have written $O(1/x)$ right away, or use it in flipped order $O(1/x) = o(1)$, but not as you have written. Please fix it. – dtldarek Oct 18 '16 at 5:01

I would write like this:

$$\frac{x}{x+1} = {\frac{\frac{x}{x}}{\frac{x}{x}+\frac{1}{x}}} \hspace{1em} \xrightarrow[x\to \infty]{} \hspace{1em} 1$$

• Or putting the $x\to\infty$ after, i.e. $\frac x{x+1} = \frac{x/x}{x/x + 1/x} \to 1, \quad x\to \infty$ as is done in many textbooks. – Therkel Oct 17 '16 at 19:54

You could just use language:

We note that $\frac{x}{x+1}=\frac{\frac{x}{x}}{\frac{x}{x}+\frac{1}{x}} \to 1$ as $x \to \infty$, since $\frac{1}{x}$ vanishes in the denominator.

Apart from the long arrow notation, mentioned by others: $$f \xrightarrow[x\to\infty]{} g$$ I would recommend, in this specific problem, to use a new variable. Note that for $$u = x+1$$ we have $$\frac x{x+1} = \frac {u-1}u = 1-\frac 1u$$ hence $$\lim_{x\to\infty} \frac x{x+1} = \lim_{u\to\infty} \left(1-\frac 1u\right) = 1-0 = 1$$