Evaluate the following limits :
`Lim_(x to 1) (x^(2) - x log x + log x - 1)/(x-1)`

Answer

To evaluate the limit

\[ \lim_{x \to 1} \frac{x^2 - x \log x + \log x - 1}{x - 1}, \]

we will follow these steps:

### Step 1: Substitute \( x = 1 \)

First, we can try substituting \( x = 1 \) directly into the expression:

\[ \frac{1^2 - 1 \cdot \log(1) + \log(1) - 1}{1 - 1} = \frac{1 - 0 + 0 - 1}{0} = \frac{0}{0}. \]

Since we get an indeterminate form \( \frac{0}{0} \), we need to simplify the expression further.

### Step 2: Simplify the Numerator

We will rewrite the numerator \( x^2 - x \log x + \log x - 1 \) to see if we can factor it or simplify it. Notice that we can group terms:

\[ x^2 - 1 - x \log x + \log x = (x^2 - 1) - (x - 1) \log x. \]

The term \( x^2 - 1 \) can be factored as \( (x - 1)(x + 1) \).

### Step 3: Factor the Numerator

Now we rewrite the expression:

\[ \frac{(x - 1)(x + 1) - (x - 1) \log x}{x - 1}. \]

We can factor out \( (x - 1) \) from the numerator:

\[ = \frac{(x - 1) \left( (x + 1) - \log x \right)}{x - 1}. \]

### Step 4: Cancel Out \( (x - 1) \)

Now we can cancel \( (x - 1) \) from the numerator and denominator (valid as long as \( x \neq 1 \)):

\[ = (x + 1) - \log x. \]

### Step 5: Take the Limit

Now we can take the limit as \( x \) approaches 1:

\[ \lim_{x \to 1} \left( (x + 1) - \log x \right). \]

Substituting \( x = 1 \):

\[ = (1 + 1) - \log(1) = 2 - 0 = 2. \]

### Conclusion

Thus, the value of the limit is

\[ \boxed{2}. \]

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