Q114, Limit of x/(x-1)-1/ln(x) as x goes to 1, L'Hospital's Rule

TL;DR

Learn how to evaluate a limit using L'Hopital's Rule when faced with an indeterminate form of infinity minus infinity.

Transcript

for point number 49 we are going to calculate a limit when X is approaching one x over X minus one minus one over Ln X this is a slightly tricky question but we will follow our strategy first plugging one into all the X for the first part I will end up with 1 over 1 minus 1 which is 0 so 1 over 0 we end up with infinity let me print down - and if y... Read More

Key Insights

• ♾️ Evaluating limits can involve various indeterminate forms, such as infinity over infinity or zero over zero.
• 💁 L'Hopital's Rule is a useful tool for evaluating limits in the defined indeterminate forms.
• 😑 Combining fractions with a common denominator can help simplify the expression and enable the application of L'Hopital's Rule.

Questions & Answers

Q: What is the initial indeterminate form encountered when evaluating the limit?

The initial indeterminate form is infinity minus infinity, which requires further work to evaluate.

Q: When can L'Hopital's Rule be applied?

L'Hopital's Rule can only be applied when the limit is in the form of zero over zero or infinity over infinity.

Q: How does combining the two fractions in the expression change the limit?

Combining the fractions creates a common denominator, which simplifies the expression and allows us to proceed with the evaluation.

Q: What is the final result of evaluating the limit using L'Hopital's Rule?

After differentiating the numerator and denominator, simplifying the expression, and plugging in the value of one, the limit is found to be 1.

Summary & Key Takeaways

• The video discusses the process of evaluating a limit when X is approaching one using L'Hopital's Rule.

• The first attempt at evaluation leads to an indeterminate form of infinity.

• In order to apply L'Hopital's Rule, the two fractions in the expression are combined by creating a common denominator.

• After simplifying the expression, plugging in the value of one into the expression leads to another indeterminate form of zero over zero.

• L'Hopital's Rule is then applied by differentiating the numerator and denominator separately.

• Finally, after simplification and plugging in the value of one, the limit is found to be 1.