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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2:07 minutes

Problem 24a

Textbook Question

Determine the following limits.

a. lim x→1^+ x / |x − 1|

Verified step by step guidance

Step 1: Understand the problem. We need to find the limit of the function \( \frac{x}{|x - 1|} \) as \( x \) approaches 1 from the right (denoted as \( x \to 1^+ \)).

Step 2: Analyze the behavior of the absolute value function. Since \( x \to 1^+ \), \( x \) is slightly greater than 1, making \( x - 1 \) positive. Therefore, \( |x - 1| = x - 1 \) in this region.

Step 3: Substitute the expression for the absolute value into the function. The function becomes \( \frac{x}{x - 1} \) for \( x > 1 \).

Step 4: Simplify the expression. As \( x \to 1^+ \), the denominator \( x - 1 \) approaches 0 from the positive side, while the numerator \( x \) approaches 1.

Step 5: Determine the behavior of the function. As \( x \to 1^+ \), \( \frac{x}{x - 1} \) tends to infinity, indicating that the limit is \( +\infty \).

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Key Concepts

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave 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 right (denoted as x→1^+).

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. In calculus, absolute values can affect the behavior of functions, especially at points where the expression inside the absolute value changes sign. For the limit in question, |x - 1| will behave differently depending on whether x is less than or greater than 1.

One-Sided Limits

One-sided limits are used to evaluate the behavior of a function as it approaches a specific point from one side only, either from the left (denoted as x→1^-) or from the right (denoted as x→1^+). This is crucial when dealing with functions that have different behaviors on either side of a point, such as discontinuities or vertical asymptotes. In this problem, we focus on the right-hand limit as x approaches 1.

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