We are tasked with finding the limit:

$\lim_{x \to 1} \left( \frac{x^x - x}{x - 1} - \log x \right)$

Let’s proceed step by step.

Step 1: Analyze $x^x$ as $x \to 1$

Recall that: $x^x = e^{x \log x}$ At $x = 1$: $x^x = e^{1 \cdot \log 1} = e^0 = 1$ Thus, $x^x - x = 1 - 1 = 0$ when $x = 1$, which suggests that the expression $\frac{x^x - x}{x - 1}$ might have an indeterminate form. We'll need to apply L'Hopital's Rule to this part.

Step 2: Apply L'Hopital’s Rule to $\frac{x^x - x}{x - 1}$

To apply L'Hopital's Rule, we differentiate the numerator and denominator.

1. Numerator derivative $f(x) = x^x - x$:

   • Derivative of $x^x$: $\frac{d}{dx} (x^x) = x^x (\log x + 1)$
   • Derivative of $-x$ is $-1$.

   So the derivative of the numerator is: $\frac{d}{dx} (x^x - x) = x^x (\log x + 1) - 1$

2. Denominator derivative $g(x) = x - 1$: The derivative of $x - 1$ is simply 1.

Therefore, applying L'Hopital's Rule: $\lim_{x \to 1} \frac{x^x - x}{x - 1} = \lim_{x \to 1} \left( x^x (\log x + 1) - 1 \right)$ At $x = 1$, this becomes: $1^1 (\log 1 + 1) - 1 = 1 \times (0 + 1) - 1 = 1 - 1 = 0$

Step 3: Handle the logarithmic term

Now, we need to handle the $-\log x$ term in the limit. As $x \to 1$, $\log x \to 0$.

Step 4: Combine the results

We found that both the terms $\frac{x^x - x}{x - 1}$ and $\log x$ tend to 0 as $x \to 1$. Therefore, the overall limit is: $\lim_{x \to 1} \left( \frac{x^x - x}{x - 1} - \log x \right) = 0 - 0 = 0$

Final Answer:

$\lim_{x \to 1} \left( \frac{x^x - x}{x - 1} - \log x \right) = 0$

Would you like more details or have any questions?

Here are 5 related questions:

1. How can we prove $x^x$ is differentiable at all points?
2. What are some other ways to compute limits involving indeterminate forms?
3. How does L'Hopital’s Rule work for higher-order indeterminate forms?
4. Can we apply Taylor series expansion around $x = 1$ to approximate the limit?
5. How does the behavior of logarithmic functions compare to power functions near singularities?

Tip: L'Hopital’s Rule is useful when both the numerator and denominator tend to 0 or infinity; always check if these conditions hold before applying.