What is a intuitive way to look at substitution? Substitution can be a very powerful tool in mathematics sometimes simplifying problems, as an example: 
$\lim\limits_{n \to \infty} {(1-\frac{1}{n})}^n = {(1+\frac{1}{n})}^{-n} = {(1+\frac{1}{n})}^{n (-1)} = e^{-1} $  leaving one with an already known expression and thus making things very easy to compute. 
As nice as substitution is, one might believe it works out any and every time. Though it does for example, looking at integrals, differential equations etc.  
But what is a good way understanding the difference between the process of substitution in both cases? It is easy to simply state the rules/theorems and then just verify them. However I find this to be very similar to most proofs using complete induction:
It's just there for verification, but not developing or finding rules. So, my question is: When is it allowed to substitute between variables, terms / equations? What would be a good way to demonstrate/explain this to freshers?
As always any constructive comment, answer or recommendation for reading is appreciated. Thanks in advance.  
 A: This question is vague, but my general advice would be to understand what the mathematical objects and symbols you're working with mean. Then you will be able to determine for yourself whether a substitution is valid. If you're just manipulating symbols with no idea of what anything you're writing down actually means, that's when you run into trouble. For example, let
$$S = 1 + 2 + 4 + 8 + \cdots.$$
Multiplying by $2$, we obtain
$$2S = 2 + 4 + 8 + 16 + \cdots$$
Subtract $S$ from $2S$ to get
$$S = 2 + 4 + 8 + 16 + \cdots - (1 + 2 + 4 + 8 + \cdots) = -1.$$
So then
$$1 + 2 + 4 + 8 + \cdots = -1,$$
which is clearly nonsensical. The symbolic manipulation all appears fine, but you've ignored the fact that $S$ is not a number because it diverges, so you can't manipulate it arithmetically as if it were a number.
A: You wrote "when is it allowed", but I think you are trying to ask something like "when is it a good idea"? 
If you are asking the latter, then I usually tell students something like
"I want to do as little work as possible". Meaning I relate the problem to something I already know, like you did in the example.
Another simple example is, I know that 
$$\int_{0}^\infty e^{-t}dt = 1.$$
Then I would ask, "Then what is
$$ \int_0^\infty \frac{1}{c}e^{-t}dt$$
for $c>0$?" Well,
\begin{align*}
\int_0^\infty \frac{1}{c}e^{-t}dt &= \frac{1}{c}\int_0^\infty c\cdot\frac{1}{c}e^{-t}dt \\
&= \frac{1}{c} \int_0^\infty e^{-t}dt \tag{1} \\
&= \frac{1}{c}
\end{align*}
since I already know that the integral in $(1)$ is equal to 1. I would then tell them to use this same technique for harder problems, to relate them to something they already know.
As for when is it allowed, that's a different story.
