Q257 / 270 · NDA 2021 (Phase 2) · Mathematics

If $f(x)=\frac{|x|}{|x|}, x \neq 0$, where [.] denotes the greatest integer function, then what is the right-hand limit of $f(x)$ at $x=1$ ?

  • A
    -1
  • B
    0
  • Correct answer
    C
    1
  • D
    Right-hand limit of $f(x)$ at $x=1$ does not exist

Solution

$f(x)=\frac{[x]}{|x|}= \begin{cases}\frac{[x]}{-x}, & x \lt 0 \\ \frac{[x]}{x} & , x \geq 0\end{cases}$ Right hand limit of $f(x)$ at $x=1$, $\mathrm{RHL}=\lim _{x \rightarrow 1} \frac{[x]}{x}=\lim _{h \rightarrow 0} \frac{[1+h]}{1+h}=1$

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

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