Definition of limit and finding a solution As creating a link of limits like
$$\lim_{a\rightarrow2b} \lim_{b\rightarrow2c} \lim_{c\rightarrow3}abc$$
is this possible? If yes, what is the formal solution steps of this question?
 A: As I said in a comment to another answer, we have a notation problem here, rather than calculation problem. Chains like that are normally evaluated right-to-left, i.e. your formula is effectively:
$$\lim_{a\to 2b}\left(\lim_{b\to 2c}\left(\lim_{c\to 3}abc\right)\right)$$
Now the innermost expression is $3ab$. Note that $c$, as a bound variable, has disappeared:
$$\lim_{a\to 2b}\left(\lim_{b\to 2c}3ab\right)$$
This means that $c$ in the "middle" limit is "some other $c$" and the result is $6ac$ with $b$ disappearing:
$$\lim_{a\to 2b}6ac$$
Now, the "outer" limit has introduced a "new" $b$ and so its value is $12bc$ with two free variables $b$ and $c$...
So, the bottom line is: the limit as you stated it is possible to calculate, but is extremely confusing. Normally, for clarity, one should not use the same letter to mean two different things anywhere near to each other, let alone in the same formula!
Note that the limit in which the order is reversed:
$$\lim_{c\to 3}\left(\lim_{b\to 2c}\left(\lim_{a\to 2b}abc\right)\right)$$
is perfectly fine and does not introduce double labelling. The innermost limit is $2b^2c$ (with $a$ disappearing), the middle limit is $8c^3$ (with $b$ disappearing) and the overall limit is then $8\cdot 3^3=216$.
