Given the function f(x) with the following conditions:
We need to determine the value of \(f(f(f(a)))=21.\)
First, let's break it down step by step:
Now, we solve for a such that \(f(f(f(a)))=21.\)
Thus, the only solution that works is for a=12.
Now, let's compute the limit:
\(\lim_{x \to 12} f(x) = f(12) = 2 \times 12 = 24\)
So, the correct answer is 144.
Let $\left\lfloor t \right\rfloor$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbb{N}$ for which
\[ \lim_{x \to 0^+} \left( x \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \dots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \dots + \left\lfloor \frac{9^2}{x^2} \right\rfloor \right) \geq 1 \]
is equal to __________.