Q196 / 270 · NDA 2016 (Phase 1) · Mathematics

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

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

Solution

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

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

Relevant Questions