a. \( \lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\eta \mu x}\right) \)

To solve the limit \( \lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\eta \mu x}\right) \), we can factor out \( \frac{1}{x} \): \[ \lim _{x \rightarrow 0}\left(\frac{1}{x} - \frac{1}{\eta \mu x}\right) = \lim _{x \rightarrow 0}\left(\frac{1}{x}\left(1 - \frac{1}{\eta \mu}\right)\right) \] This can be simplified to: \[ \lim _{x \rightarrow 0}\left(\frac{1 - \frac{1}{\eta \mu}}{x}\right) \] Now, if \( 1 - \frac{1}{\eta \mu} \neq 0 \) (which occurs unless \( \eta \mu = 1 \)), as \( x \) approaches 0, the term \( \frac{1}{x} \) will approach infinity. Thus, the limit diverges to either \( +\infty \) or \( -\infty \) depending on the sign of \( 1 - \frac{1}{\eta \mu} \): - If \( 1 - \frac{1}{\eta \mu} > 0 \Rightarrow \eta \mu < 1 \): The limit approaches \( +\infty \). - If \( 1 - \frac{1}{\eta \mu} < 0 \Rightarrow \eta \mu > 1 \): The limit approaches \( -\infty \). - If \( \eta \mu = 1 \): The limit would be \( 0 \). Hence, the final result of the limit depends on the value of \( \eta \mu \).