Key Concepts and Formulas

• Function Definition: Understanding piecewise functions and how to evaluate them based on the input's properties (even/odd).
• Composition of Functions: Evaluating $f(f(f(a)))$ by applying the function iteratively.
• Limits of Functions: Evaluating a limit as $x$ approaches a value from the left ($x \rightarrow a^-$). This involves understanding how the terms in the expression behave as $x$ gets arbitrarily close to $a$ from values less than $a$.
• Greatest Integer Function: Understanding the definition of $[t]$ (the greatest integer less than or equal to $t$) and its behavior near an integer.
• Absolute Value Function: Understanding $|x|$ and its behavior, especially when $x$ is positive.

Step-by-Step Solution

Part 1: Finding the value of 'a'

Step 1: Analyze the function $f(x)$ and the given equation $f(f(f(a))) = 21$. The function $f(x)$ is defined differently for even and odd natural numbers. We need to consider two cases for 'a': 'a' is odd, and 'a' is even.

Step 2: Case 1: Assume 'a' is odd. If 'a' is odd, then $f(a) = 2a$. Now we need to evaluate $f(f(a)) = f(2a)$. Since 'a' is odd, $2a$ is always even. Therefore, $f(2a) = (2a) - 1 = 2a - 1$. Next, we evaluate $f(f(f(a))) = f(2a - 1)$. For $f(2a-1)$ to be defined, we need to know if $2a-1$ is odd or even. Since 'a' is an integer, $2a$ is even, and $2a-1$ is always odd. Therefore, $f(2a - 1) = 2(2a - 1) = 4a - 2$. We are given $f(f(f(a))) = 21$. So, we set $4a - 2 = 21$. This gives $4a = 23$, which means $a = \frac{23}{4}$. However, the problem states that $a \in \mathbf{N}$ (natural numbers). Since $\frac{23}{4}$ is not a natural number, our assumption that 'a' is odd leads to no valid solution.

Step 3: Case 2: Assume 'a' is even. If 'a' is even, then $f(a) = a - 1$. Now we need to evaluate $f(f(a)) = f(a - 1)$. Since 'a' is even, $a - 1$ is always odd. Therefore, $f(a - 1) = 2(a - 1) = 2a - 2$. Next, we evaluate $f(f(f(a))) = f(2a - 2)$. For $f(2a-2)$ to be defined, we need to know if $2a-2$ is odd or even. Since 'a' is an integer, $2a$ is even, and $2a-2$ is always even. Therefore, $f(2a - 2) = (2a - 2) - 1 = 2a - 3$. We are given $f(f(f(a))) = 21$. So, we set $2a - 3 = 21$. This gives $2a = 24$, which means $a = 12$. Since $a=12$ is a natural number and it is even, this is a valid solution for 'a'.

Part 2: Evaluating the Limit

Step 4: Substitute the value of 'a' into the limit expression. We found $a=12$. The limit expression is $\lim\limits_{x \rightarrow \mathrm{a}^{-}}\left\{\frac{|x|^3}{\mathrm{a}}-\left[\frac{x}{\mathrm{a}}\right]\right\}$. Substituting $a=12$, we get $\lim\limits_{x \rightarrow 12^{-}}\left\{\frac{|x|^3}{12}-\left[\frac{x}{12}\right]\right\}$.

Step 5: Evaluate the limit of the first term: $\lim\limits_{x \rightarrow 12^{-}}\frac{|x|^3}{12}$. As $x \rightarrow 12^{-}$, $x$ is approaching 12 from values slightly less than 12. For values of $x$ close to 12 (and positive), $|x| = x$. So, the term becomes $\lim\limits_{x \rightarrow 12^{-}}\frac{x^3}{12}$. Since $\frac{x^3}{12}$ is a continuous function at $x=12$, we can directly substitute $x=12$: $\frac{12^3}{12} = 12^2 = 144$.

Step 6: Evaluate the limit of the second term: $\lim\limits_{x \rightarrow 12^{-}}\left[\frac{x}{12}\right]$. As $x \rightarrow 12^{-}$, $x$ takes values slightly less than 12. Let $x = 12 - \epsilon$, where $\epsilon$ is a small positive number. Then $\frac{x}{12} = \frac{12 - \epsilon}{12} = 1 - \frac{\epsilon}{12}$. Since $\epsilon$ is a small positive number, $\frac{\epsilon}{12}$ is also a small positive number. Thus, $1 - \frac{\epsilon}{12}$ is a number slightly less than 1. The greatest integer less than or equal to a number slightly less than 1 is 0. So, $\lim\limits_{x \rightarrow 12^{-}}\left[\frac{x}{12}\right] = 0$.

Step 7: Combine the results of the two limits. The original limit is the difference between the two limits we evaluated: $\lim\limits_{x \rightarrow 12^{-}}\left\{\frac{|x|^3}{12}-\left[\frac{x}{12}\right]\right\} = \lim\limits_{x \rightarrow 12^{-}}\frac{|x|^3}{12} - \lim\limits_{x \rightarrow 12^{-}}\left[\frac{x}{12}\right]$ $= 144 - 0 = 144$.

Common Mistakes & Tips

• Parity Errors: Carefully track whether the intermediate results of the function composition are odd or even. A small mistake here can lead to an incorrect value of 'a'.
• Greatest Integer Function Behavior: Remember that as $x$ approaches an integer $n$ from the left ($x \rightarrow n^-$), $[x]$ will be $n-1$. For example, $\lim\limits_{x \rightarrow 12^{-}}\left[\frac{x}{12}\right]$ is 0, not 1.
• Absolute Value: Be mindful of the domain of $x$ when dealing with $|x|$. In this case, as $x \rightarrow 12^{-}$, $x$ is positive, so $|x|=x$.

Summary

The problem requires us to first determine the value of 'a' by analyzing the given functional equation $f(f(f(a)))=21$. We consider two cases based on whether 'a' is odd or even. The case where 'a' is odd does not yield a natural number solution. The case where 'a' is even leads to $a=12$. Once 'a' is found, we evaluate the limit $\lim\limits_{x \rightarrow 12^{-}}\left\{\frac{|x|^3}{12}-\left[\frac{x}{12}\right]\right\}$. This involves evaluating the limit of each term separately. The limit of $\frac{|x|^3}{12}$ as $x \rightarrow 12^{-}$ is 144, and the limit of $\left[\frac{x}{12}\right]$ as $x \rightarrow 12^{-}$ is 0. Subtracting these results gives the final answer.

Final Answer

The final answer is $\boxed{144}$.