Let f(x)=begin cases x-1, & x\; is\; even ,\ 6pt] 2x, & x\; is\; odd ,end cases xin mathbb N . If for some ain mathbb N , fbigl(f(f(a))bigr)=21, then evaluate lim xto a - \ frac x β a - lfloor x/a rfloor \ , where lfloor trfloor denotes the greatest integer le t :

Mathematics · Limits, Continuity and Differentiability

JEE Main 2024 — 1 February, Shift 2 — Question 16

Let $f(x)=\begin{cases}x-1, & x\; \text{ is\; even},\\[6pt] 2x, & x\; \text{ is\; odd},\end{cases}\qquad x\in \mathbb{N}.$ If for some $a\in \mathbb{N}$ , $f\bigl(f(f(a))\bigr)=21,$ then evaluate $\lim_{x\to a^-}\left\{\frac{|x|^{\beta}}{a}-\left\lfloor \frac{x}{a}\right\rfloor\right\},$ where $\lfloor t\rfloor$ denotes the greatest integer $\le t$ :

Option A:
121
Option B:
144
Correct
Option C:
169
Option D:
225

Answer: B

Step-by-step solution

f(x)= \begin{cases} x-1, & x \text{\; is\; even},\\[4pt] 2x, & x \text{\; is\; odd}, \end{cases} \qquad x\in\mathbb{N}.

f\bigl(f(f(a))\bigr)=21

Case–1: If $a$ is even,

f(a)=a-1 \quad (\text{odd})

f(f(a))=2(a-1)=2a-2 \quad (\text{even})

f(f(f(a)))=(2a-2)-1=2a-3=21

\Rightarrow a=12

Case–2: If $a$ is odd,

f(a)=2a \quad (\text{even}), \qquad f(f(a))=2a-1 \quad (\text{odd})

f(f(f(a)))=2(2a-1)=4a-2=21 \quad (\text{not \;possible})

Hence,

a=12

Now compute:

\lim_{x\to 12^-}\left(\frac{|x|^{3}}{12}-\left\lfloor\frac{x}{12}\right\rfloor\right)

=\lim_{x\to 12^-} \frac{|x|^{3}}{12} \;-\; \lim_{x\to 12^-}\left\lfloor\frac{x}{12}\right\rfloor

As $x\to 12^{-}$ , $|x|^{3}\to 12^{3}=1728$ and

\left\lfloor\frac{x}{12}\right\rfloor = 0

\therefore \quad \frac{1728}{12}-0 = 144

\boxed{144}

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 1 February, Shift 2
Subject: Mathematics
Chapter: Limits, Continuity and Differentiability
Topic: Evaluation of Limit of Functions

Let f(x)= 2 x^2+5 x -3 , x in R . If m and n denote the number of points where f is not continuous and not differentiable respectively…

Answer: B

Step-by-step solution

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