\( \lim _ { x \rightarrow \frac { \pi } { 4 } } \frac { x } { x - \frac { \pi } { 4 } } \int _ { \frac { \pi } { 4 } } ^ { x } \frac { \cot ( t ) } { t } d ( t ) \)

To evaluate the limit \[ L = \lim _ { x \rightarrow \frac { \pi } { 4 } } \frac { x } { x - \frac { \pi } { 4 } } \int _ { \frac { \pi } { 4 } } ^ { x } \frac { \cot ( t ) } { t } d ( t ) \] we can recognize that as \( x \to \frac{\pi}{4} \), both the numerator \( \int _ { \frac { \pi } { 4 } } ^ { x } \frac { \cot ( t ) } { t } d(t) \) and the denominator \( x - \frac{\pi}{4} \) approach 0, resulting in a \( \frac{0}{0} \) indeterminate form. We'll use L'Hôpital's Rule, which requires us to differentiate the numerator and denominator with respect to \( x \): 1. Differentiate the numerator: Using the Fundamental Theorem of Calculus, we have \[ \frac{d}{dx} \int_{\frac{\pi}{4}}^x \frac{\cot(t)}{t} dt = \frac{\cot(x)}{x}. \] Thus, the derivative of the numerator is \[ \frac{\cot(x)}{x}. \] 2. Differentiate the denominator: The derivative of the denominator \( x - \frac{\pi}{4} \) is simply \[ 1. \] Now applying L'Hôpital's Rule gives us \[ L = \lim_{x \rightarrow \frac{\pi}{4}} \frac{x \cdot \frac{\cot(x)}{x}}{1} \] which simplifies to \[ \lim_{x \rightarrow \frac{\pi}{4}} \cot(x). \] At \( x = \frac{\pi}{4} \), we find that \[ \cot\left(\frac{\pi}{4}\right) = 1. \] Thus, the limit evaluates to \[ L = 1. \] So the final result is \[ \boxed{1}. \]