Is there a name for this observation involving single-variable limit to infinity? Obviously,

$$\lim_{n\to \infty} \sqrt{n^2 +1} = \infty$$

However, the following (I am sure) is true:

$$\lim_{n\to \infty} \sqrt{n^2 +1} = n$$

Is there a name for this kind of behavior in limits? Such as a general rule or variable specific rule?
The statement

$$\lim_{n \to \infty} \sqrt{n^2 + 1} = n$$

is incorrect and meaningless, since $n$ is varying on the left, but is left constant on the right. Perhaps you meant that

$$\lim_{n \to \infty} \sqrt{n^2 + 1} - n = 0$$

which is true. It seems that in general, what you're looking for is asymptotic analysis, or the ideas involved in Big O notation.
It should be noted that I am a physicist, not a mathematician.  Also, i do spend a good deal of time revamping notations for the higher dimensions at my site http://www.os2fan2.com .

Much of the problems seem to lie in that if a term $A$ includes meanings $a, b, c$, it is then simply because the same term is being used, that arriving at $a$, one can suppose that $c$ is also true, or alternately, because $c$ is absent, so is $a$.

A simple example of this is that zero includes the concepts of empty bag (no elephants) and empty column (no dimes in the dollars and cents).  The assumption that the Egyptians had no zero is based on the absence of a symbol for the empty column, but no mention is made of the empty bag being present.

The serious effect is the use of $\infty$ to represent any unbounded number, and then to turn around and use this as a kind of equality.  So, while $x$ and $\log(x)$ are unbounded as $x$ goes large, they are not equal.  None the less, we see the basis of the Cantor diagonal theorm is based on the notion that $x=\log(x)$ somehow comes to contention as $x$ gets large: specifically, that one can have $x$ numbers, each $x$ places long.

One is better served by replacing $\infty$ by the main approximate, so the points in a 3D lattice $p(r)=r^3$ as $r \to \infty$.  The ones that have an $r^n$ distribution are what i call `class n', are often governed by an order-n isomorphism.  Class-2 numbers are also characterisied by being able to write all members in a correctly-spelt (ie sort by spelling) series (like a decimal: decimals are a class-2 system).  Putting $\infty$ here simply prevents this from being seen.

The particular comments by rschreib and T Bongers to my comments, are based on the notion that $\lim$ must be read as a function, and that it is meaningless and wrong to use the limit control outside the side of the equal-term the limit is on.  Meaningless and wrong are two entirely different things.  They both seemed to had been able to extract the correct meaning from the equation, so what was written is meaningful.  Wrong is an issue of convention, here the artificial one that $\lim$ is a function of lower order than Addition/subtraction.  Putting it at lower than equality makes it allowable to read the RHS too.  $\pmod{b}$ is read that way, so it's not unprecedented that a symbol can be lower than an equity.

Writing something like $\sqrt{n^2+1}=n$ as $n$ goes large is perfectly meaningful.  This is exactly the meaning that OP intended, and that both of the commentators read.  Of course, one can write 'as $n$ goes large' as $\lim_{n \to \infty}$ does exactly this.  

So their justification that the statement is meaningless is more to do with supposing that the governing clause is broken by the equal sign, when set as a symbol, but is not broken when the thing is spelt out.  

I have spent a good deal of time, devising symbols for merging polytopes against the same symmetry.  What this means is that when you write down a symmetry group down, and then the decorations for each polytope to it, you can only ensure you're talking of the same orientation, when the symbols are attached to the same symmetry.

So, here for example, to write something like this, one can not assure that the counters in the two limits run to the same tune: that eg when $x=3$ on the LHS, it is equal to 3 on the RHS.  $$\lim_{x \to \infty} \sqrt{x^2+1} = \lim_{x \to \infty} x $$ 

Writing $$\lim_{x \to \infty} \sqrt{x^2+1} = x$$ can be read as if the limit governs the whole statement, and that the $x$ on the LHS and RHS are governed by the same ticks (ie if $x=3$ on the LHS, then $x=3$ on the RHS.  

It's considered bad form to use the same symbol for loop-counters and for variables outside the loop, and therefore it is apparent that the whole equation is governed by the limit.  This is why the meaning is self evident, regardless of whether it is 'right' or 'wrong'.  That is because the commentators chose to see the limit as a function governed by the equal sign, rather than governing the equal sign.  In fact, the limit governs the occurrance of its variable, wherever it falls, ie writing

$$lim_{x \to \infty} A(x)  \text{ is the same as } A(x) \; \text{ as } x \to \infty$$