Q208 / 270 · NDA 2023 (Phase 2) · Mathematics

Let $f(x)=|x|$ and $g(x)=[x]-1$, where [.] is the greatest integer function. Let $h(x)=\frac{f(g(x))}{g(f(x))}$. What is $\lim _{x \rightarrow 0-} h(x)$ equal to?

  • Correct answer
    A
    -2
  • B
    -1
  • C
    0
  • D
    2

Solution

Since, $h(x)=\frac{f(g(x))}{g(f(x))}$ $=\frac{f([x]-1)}{g(|x|)}=\frac{|[x]-1|}{[|x|]-1}$ $\begin{aligned} & \text { Since, }=\lim _{x \rightarrow 0-} h(x) \\ & =\lim _{x \rightarrow 0^{-}} \frac{|[x]-1|}{[x]-1}=\lim _{x \rightarrow 0^{-}}=\frac{|-1-1|}{-1}=-2\end{aligned}$

Hence, the correct answer is option: 'A'.

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