Understand the function $f$

The function is defined on natural numbers as

We are given $f(f(f(a)))=21$ for some $a\in \mathbb N$.

Work backwards from $21$

We need all possible preimages of $21$ under $f$.

For $f(n)=21$:

• If $n$ is even, then $f(n)=n-1=21 \Rightarrow n=22$.
• If $n$ is odd, then $f(n)=2n=21$, impossible.

So the only number mapping to $21$ is $22$.

Thus, $f(f(a))=22.$

Now solve $f(n)=22$

• If $n$ is even, then $n-1=22 \Rightarrow n=23$, but $23$ is not even, so impossible.
• If $n$ is odd, then $2n=22 \Rightarrow n=11$, and $11$ is odd, valid.

Hence, $f(a)=11.$

Now solve $f(a)=11$

• If $a$ is even, then $a-1=11 \Rightarrow a=12$, valid since $12$ is even.
• If $a$ is odd, then $2a=11$, impossible.

Therefore, $a=12.$

Evaluate the limit

We need

with $a=12$.

Since $x\to 12^-$ and $12>0$, for $x$ near $12$ we have $|x|=x$.

Also, when $x\to 12^-$,

Therefore,

for all $x<12$ sufficiently close to $12$.

Hence the limit becomes

Match with options

$144$ corresponds to Option D.