Lt (x→1) (x^n - n) / (x - 1 - log(x))

Steps to Solve

1. Direct Substitution

Start by substituting ( x = 1 ) into the expression.

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

This results in an indeterminate form of type ( \frac{0}{0} ).

2. Apply L'Hôpital's Rule

Since we have the indeterminate form ( \frac{0}{0} ), we can apply L'Hôpital's Rule, which states we can take the derivative of the numerator and the derivative of the denominator.

3. Differentiate Numerator and Denominator

Differentiate the numerator ( x^n - n ):

$$ \frac{d}{dx}(x^n - n) = nx^{n-1} $$

Differentiate the denominator ( x - 1 - \log(x) ):

$$ \frac{d}{dx}(x - 1 - \log(x)) = 1 - \frac{1}{x} $$

4. Substitute Again

Now substitute ( x = 1 ) again after differentiating:

Numerator:

$$ n(1)^{n-1} = n $$

Denominator:

$$ 1 - \frac{1}{1} = 0 $$

Again, we have ( \frac{n}{0} ). Thus, we should apply L'Hôpital's Rule one more time.

5. Differentiate Again

Differentiate the new numerator ( nx^{n-1} ):

$$ \frac{d}{dx}(nx^{n-1}) = n(n-1)x^{n-2} $$

Differentiate the new denominator ( 1 - \frac{1}{x} ):

$$ \frac{d}{dx}(1 - \frac{1}{x}) = \frac{1}{x^2} $$

6. Final Substitution

Substituting ( x = 1 ) again:

Numerator:

$$ n(n-1)(1)^{n-2} = n(n-1) $$

Denominator:

$$ \frac{1}{1^2} = 1 $$

Thus,

$$ \lim_{x \to 1} \frac{nx^{n-1}}{1 - \frac{1}{x}} = n(n-1) $$