# 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?

## 1 Answer

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 $$8\cdot 3^3=216$$.

• "This is possible to do, but is extremely confusing. Normally, for clarity, you should not use the same letter to mean two different things anywhere near to each other, let alone in the same formula!" -- I would go even further and say that doing this should be avoided at (practically) all costs. @OP, if you wanted to express the limit matching Stinking Bishop's interpretation (the only reasonable interpretation), then I would write$$\lim_{a \to 2b} \lim_{x \to 2c} \lim_{y \to 3} axy.$$ – Theo Bendit Feb 28 at 9:43