Let f(x)= gathered x-1,x 2x,, x gatheredx N.

Question

Let $f(x)= \begin{gathered} x-1,x \ 2x,\, x \end{gathered}x\in N$.

If for some $a\in N,f(f(f(a)))=21$, then $\underset{x	o {a}^{-}}{lim}{\frac{|x{|}^{3}}{a}-[\frac{x}{a}]}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is equal to :

Accepted Answer

$f(x)= \begin{gathered} x-1,x \ 2x,\, x \end{gathered}x\in N$

Let $a$ is odd

$\begin{gathered} \Rightarrow f(a)=2a \ \Rightarrow f(f(a))=2a-1 \ \Rightarrow f(f(f(a)))=2(2a-1) \end{gathered}$

$2(2a-1)=21$ Not possible for any $a\in N$

Let $a$ is even

$\begin{gathered} \Rightarrow f(a)=a-1 \ \Rightarrow f(f(a))=2(a-1) \ \Rightarrow f(f(f(a)))=2(a-1)-1=2a-3 \ 2a-3=21\, \Rightarrow a=12 \end{gathered}$

Now

$\begin{gathered} \underset{x	o {12}^{-}}{lim}(\frac{|x{|}^{3}}{2}-[\frac{x}{12}]) \ =\underset{x	o {12}^{-}}{lim}\frac{|x{|}^{3}}{12}-\underset{x	o {12}^{-}}{lim}[\frac{x}{12}] \ =144-0=144. \end{gathered}$

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