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

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1. We recognize the given limit
   $$
   \lim_{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{*} \ln x_{k}^{*} \Delta x_{k}
   $$
   as a Riemann sum. In a typical Riemann sum of the form
   $$
   \lim_{\Delta \rightarrow 0} \sum_{k=1}^{n} f(x_k^*) \Delta x_{k},
   $$
   the summand $$ f(x_k^*) $$ corresponds to the function evaluated at the sample point $$ x_k^* $$.

2. In our case, the summand is
   $$
   x_{k}^{*} \ln x_{k}^{*},
   $$
   which implies that the function is
   $$
   f(x)=x\ln x.
   $$

3. Since the Riemann sum is taken on the interval $$[1,2]$$, the limit is equivalent to the definite integral of $$ f(x) $$ from 1 to 2. Hence, we can express the limit as:
   $$
   \int_{1}^{2} x\ln x\,dx.
   $$
The function is $$ f(x) = x \ln x $$, and the limit can be expressed as the definite integral: $$ \int_{1}^{2} x \ln x \, dx $$

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

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