Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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2:24 minutes

Problem 51

Textbook Question

Determine the following limits.
lim x→1^− x/ ln x

Verified step by step guidance

Identify the type of limit: This is a one-sided limit as \( x \to 1^- \), meaning we approach 1 from the left.

Substitute \( x = 1 \) into the expression \( \frac{x}{\ln x} \) to check if it results in an indeterminate form.

Recognize that as \( x \to 1^- \), \( \ln x \to \ln(1) = 0 \) and \( x \to 1 \), leading to the indeterminate form \( \frac{1}{0} \).

Apply L'Hôpital's Rule, which is used to evaluate limits of indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Differentiate the numerator and the denominator separately.

Differentiate: The derivative of the numerator \( x \) is 1, and the derivative of the denominator \( \ln x \) is \( \frac{1}{x} \). Rewrite the limit as \( \lim_{x \to 1^-} \frac{1}{\frac{1}{x}} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or infinity. In this case, we are interested in the limit as x approaches 1 from the left, which requires analyzing the function's values as they get closer to this point.

Natural Logarithm (ln)

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a crucial function in calculus, particularly in limits and derivatives, as it has unique properties, such as being undefined for non-positive values. Understanding how ln(x) behaves as x approaches 1 is essential for evaluating the limit in the given question.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In the limit lim x→1^− x/ln x, both the numerator and denominator approach 0 as x approaches 1 from the left, creating an indeterminate form. Recognizing this allows us to apply techniques such as L'Hôpital's Rule to resolve the limit.