Consider the following limit of Riemann sums of a function \( f \) on \( [a, b] \). Identify \( f \) and express the limit as a definite integral. \( \lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{*} \tan x_{k}^{*} \Delta x_{k} ;[2,3] \) The limit, expressed as a definite integral, is

To identify the function \( f \) and express the limit as a definite integral, we start by analyzing the given Riemann sum: \[ \lim_{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{*} \tan x_{k}^{*} \Delta x_{k}. \] Here, \( x_{k}^{*} \) represents sample points chosen from subintervals in \( [2, 3] \), and \( \Delta x_{k} \) is the width of these subintervals. The expression \( x_{k}^{*} \tan x_{k}^{*} \) suggests that this is a Riemann sum for the function \( f(x) = x \tan x \). As the limit approaches zero, this sum becomes the definite integral of \( f \) over the interval \( [2, 3] \). Therefore, we can express the limit as a definite integral: \[ \int_{2}^{3} x \tan x \, dx. \] Thus, the limit expressed as a definite integral is \[ \int_{2}^{3} x \tan x \, dx. \]