Limits, Continuity & Differentiability
Floor Function Limit via Piecewise Function Composition
nta_pyq_2024_jan
Grade 12
Question:
Let $f(x)=\begin{cases}x-1,&x\text{ is even}\\2x,&x\text{ is odd}\end{cases}$, $x\in\mathbb{Z}$. If for some $a\in\mathbb{N}$, $f(f(f(a)))=21$, then $\displaystyle\lim_{x\to a^-}\left\{\dfrac{|x|^3}{a}-\left[\dfrac{x}{a}\right]\right\}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is equal to:
Step-by-Step Solution
Key Concept: Find $a\in\mathbb{N}$ such that $f(f(f(a)))=21$. Case 1: $a$ even $\Rightarrow f(a)=a-1$ (odd) $\Rightarrow f(f(a))=2(a-1)$ (even) $\Rightarrow f(f(f(a)))=2a-3=21\Rightarrow a=12$. Case 2: $a$ odd gives no natural number solution. So $a=12$. Compute $\lim_{x\to12^-}\left(\frac{|x|^3}{12}-\left[\frac{x}{12}\right]\right)$.
$f(f(f(a)))=21\Rightarrow a=12$. $\lim_{x\to12^-}\left(\frac{x^3}{12}-\left[\frac{x}{12}\right]\right)=144-0=144$.
Correct Answer: 2