Question

$\underset{x\to 0^{-}}{lim}{\left(\ln \left(\right\{x\}+|[x\left]|\right)\right)}^{\left\{x\right\}}$, where $\{\cdot \}$ denotes fractional part function and $[\cdot ]$ denotes greatest integer function, is

• A. $0$
• B. $1$
• C. $\ln 2$
• D. $\ln \frac{1}{2}$

Answer 1

Answer: (C)

$\ln 2$

$\underset{x\to 0^{-}}{lim}{\left(\ln \left(\right\{x\}+|[x\left]|\right)\right)}^{\left\{x\right\}}$

Substitute $x$ as $0-h$, where $h\to 0$

$\Rightarrow \underset{h\to 0}{lim}{\left(\ln \left(\right\{0-h\}+|[0-h\left]|\right)\right)}^{\{0-h\}}$

$\Rightarrow \underset{h\to 0}{lim}{\left(\ln (1-h+1)\right)}^{1-h}$

$\Rightarrow \underset{h\to 0}{lim}{\left(\ln (2-h)\right)}^{1-h}$

$\Rightarrow \ln 2$