How to Evaluate Limits of Logarithmic Functions

TL;DR

To evaluate the limit of ln(x) - 1 over x - e as x approaches e, use L'Hopital's Rule since direct substitution gives an indeterminate form. Taking derivatives leads to the limit simplifying to 1/e, confirming that the limit is approximately 0.368 when evaluated with values close to e.

Transcript

consider this problem what is the limit as x approaches e of the expression ln x minus 1 over x minus e how can we evaluate this logarithmic limit well we can try direct substitution we can replace x with e so we're going to have l and e minus one divided by e minus e the natural log of e is one and so one minus one is zero e minus e is also zero s... Read More

Key Insights

• 💦 Evaluating logarithmic limits can be tricky, as direct substitution does not always work.
• 💁 L'Hopital's Rule provides an effective method for evaluating indeterminate forms.
• ⛔ The rule states that the limit of f(x)/g(x) is equal to the limit of f'(x)/g'(x) when both derivatives exist.
• 😑 Applying L'Hopital's Rule to the given logarithmic limit simplifies the expression and gives the correct answer of 1/e.
• 📫 To confirm the answer, decimal values obtained by plugging in x values close to e can be compared to the decimal value of 1/e.
• ♾️ L'Hopital's Rule is particularly useful for indeterminate forms of zero over zero or infinity over infinity.
• 😑 Understanding L'Hopital's Rule is crucial for solving complex limits involving logarithmic expressions.

Questions & Answers

Q: How can we evaluate the limit of ln(x) - 1 / (x - e) as x approaches e?

Direct substitution fails because it results in an indeterminate form. To solve this, we can use L'Hopital's Rule. By taking the derivative of ln(x) and (x - e), we can simplify the expression to 1/x. Substituting x with e gives us the final answer of 1/e.

Q: What is L'Hopital's Rule and when is it used?

L'Hopital's Rule states that for an indeterminate form of a limit, f(x)/g(x), if both f'(x) and g'(x) exist, the limit is equal to the limit of f'(x)/g'(x). It is used when direct substitution fails or results in an indeterminate form.

Q: How do we confirm if our answer is correct?

To confirm the answer, we can compare the decimal value of 1/e (approximately 0.3678794412) with the value obtained by plugging in x values close to e. As we get closer to e, the expression approaches the decimal value, validating our answer.

Q: What happens if we have an indeterminate form of 0/0 or infinity/infinity?

In such cases, L'Hopital's Rule can be used to evaluate the limit. By taking the derivative of both the numerator and denominator, we can simplify the expression and find the limit.

Summary & Key Takeaways

• The video discusses how to evaluate the limit of ln(x) - 1 / (x - e) as x approaches e.

• Direct substitution fails to solve the indeterminate form of the limit, so L'Hopital's Rule is used.

• L'Hopital's Rule states that for a limit of f(x)/g(x), the limit is equal to f'(x)/g'(x).

• Applying L'Hopital's Rule to the given problem, the derivative of ln(x) is 1/x, leading to the final answer of 1/e.