Evaluate lim x->1 (x^2 - x) / (x - 1 - log(x))

Steps to Solve

1. Substituting x = 1

Start by substituting $x = 1$ into the expression to check for an indeterminate form:

$$ \frac{(1^2 - 1)}{(1 - 1 - \log(1)) = \frac{0}{0} $$

Since this results in the form $\frac{0}{0}$, we need to use L'Hôpital's rule.

2. Applying L'Hôpital's Rule

Differentiate the numerator and the denominator separately:

• The derivative of the numerator ($x^2 - x$) is $2x - 1$.
• The derivative of the denominator ($x - 1 - \log(x)$) is $1 - \frac{1}{x}$.

Now we can rewrite the limit as:

$$ \lim_{x \to 1} \frac{2x - 1}{1 - \frac{1}{x}} $$

3. Finding the new limit

Substitute $x = 1$ into the new expression:

$$ \frac{2(1) - 1}{1 - 1} = \frac{1}{0} $$

This is undefined, so we need to analyze the limit further.

4. Further Applying L'Hôpital's Rule

Since we again have an indeterminate form, apply L'Hôpital's rule once more. Differentiate the numerator and denominator again:

• The derivative of the numerator ($2x - 1$) is $2$.
• The derivative of the denominator ($1 - \frac{1}{x}$) is $\frac{1}{x^2}$.

Now we have:

$$ \lim_{x \to 1} \frac{2}{\frac{1}{x^2}} = \lim_{x \to 1} 2x^2 = 2(1^2) = 2 $$

5. Conclusion

The limit as $x$ approaches 1 is $2$.