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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2:01 minutes

Problem 2.2.70b

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.

Verified step by step guidance

First, understand the function h(x) = (x² − 2x − 3)/(x² − 4x + 3). This is a rational function, which means it is the ratio of two polynomials.

Identify any points of discontinuity by setting the denominator equal to zero: x² − 4x + 3 = 0. Solve this quadratic equation to find the values of x that make the denominator zero.

Factor the denominator: x² − 4x + 3 = (x - 3)(x - 1). The function h(x) is undefined at x = 3 and x = 1, which are the points of discontinuity.

To estimate the limit as x approaches 3, use a graphing calculator to plot the function h(x). Use the Zoom feature to focus on the area around x = 3.

Utilize the Trace feature on the graphing calculator to observe the behavior of the y-values as x approaches 3 from both sides. This will help you estimate the limit of h(x) as x approaches 3.

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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 in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, understanding limits helps in analyzing the behavior of the function h(x) as x approaches 3, which is crucial for determining continuity and potential discontinuities in the function.

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate plane, which visually represents its behavior. In this exercise, using a graphing calculator to visualize h(x) near x = 3 allows for a better understanding of how the function behaves around that point, aiding in the estimation of limits.

Zoom and Trace Features

The Zoom and Trace features on a graphing calculator allow users to adjust the viewing window and track the values of a function as the input changes. This is particularly useful for estimating y-values as x approaches a specific point, such as 3 in this case, providing a practical method to support conclusions drawn from limit analysis.

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Related Practice

Textbook Question

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 (−1) / (x² (x + 1))

Textbook Question

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

b. lim x→0⁻ 2 / (3x¹/³)

Textbook Question

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 4 / x²/⁵

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 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

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.

Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx →4 (9 − x) = 5

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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.

Key Concepts

Limits

Graphing Functions

Zoom and Trace Features

Watch next

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.