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

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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1:51 minutes

Problem 2.2.71b

Textbook Question

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let f(x)=(x² − 1)/(|x| − 1).

b. Support your conclusion in part (a) by graphing f near c = -1 and using Zoom and Trace to estimate y-values on the graph as x→−1.

Verified step by step guidance

First, understand the function f(x) = (x² − 1)/(|x| − 1). This function involves an absolute value in the denominator, which can affect the behavior of the function around x = -1.

To analyze the limit as x approaches -1, consider the behavior of the function from both sides of x = -1. This means evaluating the function as x approaches -1 from the left (x → -1⁻) and from the right (x → -1⁺).

Graph the function using a graphing calculator. Set the window to focus on values of x near -1, such as from -1.5 to -0.5, to observe the behavior of the function around this point.

Use the Zoom feature to get a closer look at the graph near x = -1. This will help you see how the function behaves as x approaches -1 from both sides.

Utilize the Trace feature on the graphing calculator to estimate the y-values of the function as x approaches -1. This will provide a visual approximation of the limit by observing the values that f(x) approaches as x gets closer to -1.

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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 where they are not defined. For example, evaluating the limit of f(x) as x approaches -1 allows us to determine the function's value or behavior at that point.

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate plane to visualize its behavior. This technique is essential for understanding limits, as it allows us to see how the function behaves as it approaches a specific x-value. Using tools like graphing calculators can enhance this process by providing precise visual representations and enabling the estimation of y-values as x approaches a limit.

Continuity and Discontinuity

Continuity refers to a function being unbroken and having no gaps at a point, meaning the limit at that point equals the function's value. Discontinuity occurs when a function has a break, jump, or point where it is not defined. Understanding whether f(x) is continuous or discontinuous at x = -1 is crucial for accurately estimating limits and interpreting the function's behavior near that point.

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Related Practice

Textbook Question

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let G(x)=(x + 6)/(x² + 4x − 12)

b. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as x→−6.

Textbook Question

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let h(x)=(x² − 2x − 3)/(x² − 4x + 3)

b. Support your conclusions in part (a) by graphing h near c = 3 and using Zoom and Trace to estimate y-values on the graph as x→3.

Textbook Question

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let F(x)=(x² + 3x + 2)/(2−|x|)

b. Support your conclusion in part (a) by graphing F near c = -2 and using Zoom and Trace to estimate y-values on the graph as x→−2.

Textbook Question

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let g(θ) = (sinθ) / θ.

b. Support your conclusion in part (a) by graphing g near θ₀ = 0.

Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx →4 (9 − x) = 5

Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→3 (3x − 7) = 2

Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→9 √(x − 5) = 2

Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 √(4 − x) = 2

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Estimating Limits[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.Let f(x)=(x² − 1)/(|x| − 1).b. Support your conclusion in part (a) by graphing f near c = -1 and using Zoom and Trace to estimate y-values on the graph as x→−1.

Key Concepts

Limits

Graphing Functions

Continuity and Discontinuity

Watch next

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let f(x)=(x² − 1)/(|x| − 1).

b. Support your conclusion in part (a) by graphing f near c = -1 and using Zoom and Trace to estimate y-values on the graph as x→−1.