Q238 / 270 · NDA 2024 (Phase 1) · Mathematics

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

  • Correct answer
    A
    $-\frac{3}{2}$
  • B
    $-\frac{1}{2}$
  • C
    $\frac{1}{2}$
  • D
    $\frac{3}{2}$

Solution

and $\begin{aligned} f(x) & =|x|+1 \\ g(x) & =[x]-1 \\ h(x) & =\frac{f(x)}{g(x)}=\frac{|x|+1}{[x]-1}\end{aligned}$ $\lim _{x \rightarrow 0} h(x)+\lim _{x \rightarrow 0^{+}} h(x)=\lim _{x \rightarrow 0^{-}} \frac{|x|+1}{[x]-1}$ $\begin{array}{ll}=\lim _{x \rightarrow 0^{-}} \frac{|x|+1}{[x]-1} & \\ =\frac{0+1}{-1-1}+\frac{0+1}{0-1} & \left\{\begin{array}{l}{[0-h]=-1} \\ {[0+h]=0}\end{array}\right\} \\ =\frac{1}{2}-1=-\frac{3}{2} & \end{array}$

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

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