In limits, how is change of variables defined?

Question

When taking limits, for example as $n \rightarrow \infty$, it is obvious that $f(n)$ and $f(n + 1/2)$ approach the same limit. In both cases, you are essentially taking $f$ at higher and higher values, so the result should be the same.

But what is the formal justification of this rule?

EDIT:

Also, there are situations such as $n \rightarrow \infty$ is identical to $k \rightarrow 0$ in $1/k$. Thus, the limit of $f(n)$ is the same as the limit of $f(1/k)$, but, again, how to show it formally?

Answer 1

$$ \lim\limits_{n\rightarrow\infty}x_n=x \iff (\forall \,\varepsilon>0 \,\exists N\in\mathbb{N} : n>N \implies|x_n-x|<\varepsilon)$$

Thus you can see that if $n>N$, then $n+\frac{1}{2}>N$ too. Similarly $2n$ or some function of $n$ under certain conditions.

EDIT: obviously since $N$ is arbitrary, things like $\frac{n}{2}$ hold too because $2N$ is just as good a constant. And to answer to your edit, it's enough to prove that $\lim\limits_{n\rightarrow\infty}\frac{1}{n}=0$ and $\lim\limits_{k\rightarrow0}\frac{1}{k}=\infty$

EDIT2: This answer is ultimately not correct. See the comments for Clement C. 's counterexample.

Answer 2

"Obvious" or not, it is not always true that $f(n)$ and $f(n+1/2)$ have the same limit. Consider $f(x) = \sin^2(\pi x)$ for example.

So it's impossible to say how to prove it formally; in cases where it is true the proof depends on what function $f$ you're talking about.

Answer 3

Regarding your edit, one can use the definition of limit to show that if $\lim_{x\to a}f(x)=L$ and $\lim_{x\to b}g(x)=a$, then $$ \lim_{x\to b} f(g(x))=L. $$