# Thread: Limit Problem - # 3

1. ## Limit Problem - # 3

$\lim x \rightarrow 0^{+}[\sin x \ln 2x]$

$\lim x \rightarrow 0^{+}[\sin (0) \ln 2(0)]$

$\lim x \rightarrow 0^{+}[(0)( \ln 0)]$

$\lim x \rightarrow 0^{+}[(0)( -\infty)]$ Indeterminate

$\lim x \rightarrow 0^{+}[\dfrac{\ln2 x}{\dfrac{1}{\sin x}}]$ Rewrite

$\lim x \rightarrow 0^{+}[\dfrac{\ln2 x}{\csc x}]$

$\lim x \rightarrow 0^{+}[\dfrac{\dfrac{d}{dx} \ln x}{\dfrac{d}{dx} \csc x}]$

$\lim x \rightarrow 0^{+}[\dfrac{\dfrac{1}{x}}{-\csc x \cot x}]$

$\lim x \rightarrow 0^{+}[\dfrac{\dfrac{1}{0}}{-\csc 0 \cot 0}]$

2. ## Re: Limit Problem - # 3

Originally Posted by Jason76
$\lim x \rightarrow 0^{+}[\sin x \ln 2x]$

$\lim x \rightarrow 0^{+}[\sin (0) \ln 2(0)]$

$\lim x \rightarrow 0^{+}[(0)( \ln 0)]$

$\lim x \rightarrow 0^{+}[(0)( -\infty)]$ Indeterminate

$\lim x \rightarrow 0^{+}[\dfrac{\ln2 x}{\dfrac{1}{\sin x}}]$ Rewrite

$\lim x \rightarrow 0^{+}[\dfrac{\ln2 x}{\csc x}]$

$\lim x \rightarrow 0^{+}[\dfrac{\dfrac{d}{dx} \ln x}{\dfrac{d}{dx} \csc x}]$

$\lim x \rightarrow 0^{+}[\dfrac{\dfrac{1}{x}}{-\csc x \cot x}]$

$\lim x \rightarrow 0^{+}[\dfrac{\dfrac{1}{0}}{-\csc 0 \cot 0}]$
\displaystyle \begin{align*} \sin{(x)}\ln{(2x)} &= \frac{\sin{(x)}}{\left[ \ln{(2x)} \right] ^{-1} } \end{align*}

This is now of the form \displaystyle \begin{align*} \frac{0}{0} \end{align*} so L'Hospital's Rule can be applied.