- #1

Eclair_de_XII

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- 91

- Homework Statement
- Define ##f:(-1,0)\rightarrow \mathbb{R}## by ##f(x)=-\ln(-x)##. Show that ##f## is unbounded.

- Relevant Equations
- A function ##f## is said to be unbounded if for all positive numbers ##M##, there is a ##y## in ##\textrm{dom}(f)## such that ##|f(y)|>M##.

So far, I found the derivative of ##f##:

\begin{align*}

\frac{d}{dx}\,f(x)&=&-\frac{d}{dx}\,\ln(-x)\\

&=&-\left(\frac{1}{(-x)}\right)(-1)\\

&=&-\frac{1}{x}

\end{align*}

##f'(x)## is always positive and never zero on its domain.

Hence, ##f## does not have a local maximum and is always increasing on the interval ##(-1,0)##.

Are these conditions sufficient to argue that ##\ln## is unbounded near zero?

\begin{align*}

\frac{d}{dx}\,f(x)&=&-\frac{d}{dx}\,\ln(-x)\\

&=&-\left(\frac{1}{(-x)}\right)(-1)\\

&=&-\frac{1}{x}

\end{align*}

##f'(x)## is always positive and never zero on its domain.

Hence, ##f## does not have a local maximum and is always increasing on the interval ##(-1,0)##.

Are these conditions sufficient to argue that ##\ln## is unbounded near zero?