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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2:26 minutes

Problem 24c

Textbook Question

Determine the following limits.

c. lim x→1 x / |x − 1|

Verified step by step guidance

Step 1: Understand the nature of the absolute value function. The expression |x - 1| represents the distance of x from 1 on the number line. It is defined as |x - 1| = x - 1 if x >= 1 and |x - 1| = -(x - 1) if x < 1.

Step 2: Consider the limit from the right (x approaches 1 from values greater than 1). In this case, |x - 1| = x - 1, so the expression becomes x / (x - 1).

Step 3: Evaluate the limit from the right. As x approaches 1 from the right, the denominator (x - 1) approaches 0, causing the expression x / (x - 1) to approach infinity. Therefore, the right-hand limit is positive infinity.

Step 4: Consider the limit from the left (x approaches 1 from values less than 1). In this case, |x - 1| = -(x - 1), so the expression becomes x / (-(x - 1)) = -x / (x - 1).

Step 5: Evaluate the limit from the left. As x approaches 1 from the left, the denominator (x - 1) approaches 0, causing the expression -x / (x - 1) to approach negative infinity. Therefore, the left-hand limit is negative infinity.

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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. Evaluating limits is essential for defining derivatives and integrals, which are core components of calculus.

Absolute Value Function

The absolute value function, denoted as |x|, measures the distance of a number from zero on the number line, always yielding a non-negative result. In the context of limits, the absolute value can affect the behavior of a function as it approaches a point, particularly when the function crosses zero, leading to different left-hand and right-hand limits.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either the left (denoted as lim x→c-) or the right (denoted as lim x→c+). In the given limit problem, evaluating one-sided limits is crucial because the absolute value function creates a piecewise scenario that can lead to different outcomes depending on the direction of approach.

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