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

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Understand the Problem

The question involves evaluating the limit as x approaches 1 of the expression (x^2 - x) / (x - 1 - log(x)). It requires understanding the concepts of limits and derivatives.

Answer

The limit is $2$.
Answer for screen readers

The limit is $2$.

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.

  1. 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:

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

  1. 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.

  1. 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:

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

  1. Conclusion

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

The limit is $2$.

More Information

L'Hôpital's rule is a powerful tool for evaluating limits that result in indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$. By repeatedly differentiating, we can often find a limit that is easier to evaluate.

Tips

  • Forgetting to apply L'Hôpital's rule correctly; it's important to differentiate both the numerator and denominator.
  • Not recognizing when the expression still leads to an indeterminate form after the first application of L'Hôpital's rule.

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