What is \lim_{x\rightarrow 0^{-}}h(x)+\lim_{x\rightarrow 0^{+}}h(x) equal to ?

Consider the following for the next two (02) items that follow : Let f(x)=|x|+1 and g(x)=[x]-1, where [.] is the greatest integer function. Let h(x)=\frac{f(x)}{g(x)}

  1. A. -\frac{3}{2} ✓
  2. B. -\frac{1}{2}
  3. C. \frac{1}{2}
  4. D. \frac{3}{2}

Correct Answer: A. -\frac{3}{2}

Explanation

We need the left and right limits of h(x) = \frac{|x|+1}{[x]-1}. As x \to 0^-, |x| \to 0 and [x] = -1, so h(x) \to \frac{0+1}{-1-1} = -\frac{1}{2}. As x \to 0^+, |x| \to 0 and [x] = 0, so h(x) \to \frac{0+1}{0-1} = -1. The sum of these limits is -\frac{1}{2} + (-1) = -\frac{3}{2}.

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