2 Starting the Analysis

Since the latter two cases are undefined (the natural log is not defined at zero) and we are only interested in real numbers (i.e., $\mathbb{R}$)… our focus will be on the first two cases.

$$ \begin{align} (\text{cond }f)(x) &= \begin{cases} \left| \frac{xf^{\prime}(x)}{f(x)} \right| & \text{if } x \ne 0 \land x \in \mathbb{R}_{+}\\ \left| xf^{\prime}(x) \right| & \text{if } x = 1 \\ \end{cases} \end{align} $$

The first case leads to two separate cases…

$$ \begin{align} (\text{cond }f)(x) &= \begin{cases} \left| \frac{xf^{\prime}(x)}{f(x)} \right| & x \to \infty\\ \left| \frac{xf^{\prime}(x)}{f(x)} \right| & x \to 0\\ \left| xf^{\prime}(x) \right| & x = 1 \\ \end{cases} \end{align} $$

We also need to analyze $x \to 1$.

$$ \begin{align} (\text{cond }f)(x) &= \begin{cases} \left| \frac{xf^{\prime}(x)}{f(x)} \right| & x \to \infty\\ \left| \frac{xf^{\prime}(x)}{f(x)} \right| & x \to 0\\ \left| \frac{xf^{\prime}(x)}{f(x)} \right| & x \to 1\\ \left| xf^{\prime}(x) \right| & x = 1 \\ \end{cases} \end{align} $$

3 The Derivative and Simplification

We know $f^{\prime}(x) = \frac{1}{x}$ by the definition of the derivative of the natural log. We can now evaluate $|xf^{\prime}(x)|$…

$$ \left|xf^{\prime}(x)\right| = \left|x\frac{1}{x}\right| = 1 $$

If we plug that result and $f(x)$ into four (4) cases…

$$ \begin{align} (\text{cond }f)(x) &= \begin{cases} \left| \frac{1}{\ln(x)} \right| & x \to \infty\\ \left| \frac{1}{\ln(x)} \right| & x \to 0\\ \left| \frac{1}{\ln(x)} \right| & x \to 1\\ 1 & x = 1 \\ \end{cases} \end{align} $$

We now have three (3) cases left to evaluate. Since we have $\frac{1}{\ln(x)}$… we want to start with the behavior of $\ln(x)$.

$$ \begin{eqnarray} \lim_{x \to \infty}\ln(x) \to \infty &\implies& \left|\lim_{x \to \infty }\frac{1}{\ln(x)}\right| \to 0 \\ \lim_{x \to 1}\ln(x) \to 0 &\implies& \left|\lim_{x \to 1 }\frac{1}{\ln(x)}\right| \to \infty \\ \lim_{x \to 0}\ln(x) \to -\infty &\implies& \left|\lim_{x \to 0 }\frac{1}{\ln(x)}\right| \to 0 \\ \end{eqnarray} $$

Using these results…

$$ \begin{align} (\text{cond }f)(x) &= \begin{cases} \text{Well Conditioned} & x \to \infty\\ \text{Well Conditioned} & x \to 0\\ \text{Ill Conditioned} & x \to 1\\ \text{Well Conditioned} & x = 1 \\ \end{cases} \end{align} $$