Q197 / 270 · NDA 2016 (Phase 1) · Mathematics

Consider the function $\mathrm{f}(\mathrm{x})=\frac{\mathrm{a}^{[\mathrm{x}]+\mathrm{x}}-1}{[\mathrm{x}]+\mathrm{x}}$ where $[\cdot]$ denotes the greatest integer function. What is $\lim _{x \rightarrow 0^{-}} f(x)$ equal to?

  • A
    0
  • B
    ln a
  • Correct answer
    C
    $1-\mathrm{a}^{-1}$
  • D
    Limit does not exist

Solution

$\begin{aligned} \lim _{x \rightarrow 0^{-}} f(x) & =\lim _{h \rightarrow 0}\left[\frac{a^{[0-h]+(0-h)}-1}{[0-h]+(0-h)}\right] \\ & =\lim _{h \rightarrow 0} \frac{a^{[-h]-h}-1}{[-h]+(-h)} \\ & =\lim _{h \rightarrow 0} \frac{a^{-1-h}-1}{-1-h} \\ & =\frac{a^{-1-0}-1}{-1-0}=\frac{a^{-1}-1}{-1} \\ & =\lim _{h \rightarrow 0^{-}} f(x)=\left(1-a^{-1}\right)\end{aligned}$

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

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