lim _(x arrow 202 u)-((1)/(x-202 u)-(1)/(|x-202 u|))

To evaluate the limit \( \lim_{x \rightarrow 202u} -\left(\frac{1}{x-202u} - \frac{1}{|x-202u|}\right) \), let's analyze the expression inside the limit.<br /><br />First, consider the expression \( \frac{1}{x-202u} - \frac{1}{|x-202u|} \).<br /><br />When \( x \) approaches \( 202u \) from the right (i.e., \( x \to 202u^+ \)), \( x - 202u \) is positive, so \( |x - 202u| = x - 202u \). Thus, the expression becomes:<br />\[ \frac{1}{x-202u} - \frac{1}{x-202u} = 0 \]<br /><br />When \( x \) approaches \( 202u \) from the left (i.e., \( x \to 202u^- \)), \( x - 202u \) is negative, so \( |x - 202u| = -(x - 202u) \). Thus, the expression becomes:<br />\[ \frac{1}{x-202u} - \frac{1}{-(x-202u)} = \frac{1}{x-202u} + \frac{1}{x-202u} = \frac{2}{x-202u} \]<br /><br />Now, we need to evaluate the limit of the negative of this expression as \( x \) approaches \( 202u \):<br />\[ \lim_{x \to 202u} -\left(\frac{1}{x-202u} - \frac{1}{|x-202u|}\right) \]<br /><br />From the above analysis, we see that:<br />- When \( x \to 202u^+ \), the expression is \( 0 \).<br />- When \( x \to 202u^- \), the expression is \( -\frac{2}{x-202u} \).<br /><br />Since the limit from the left and the right do not match, the limit does not exist in this case. Therefore, the limit is undefined.