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

Previous problemNext problem

1:59 minutes

Problem 2.2.69b

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.

Verified step by step guidance

First, identify the function G(x) = (x + 6)/(x² + 4x − 12). Notice that the denominator can be factored to find any points of discontinuity.

Factor the denominator: x² + 4x − 12 = (x + 6)(x − 2). This reveals that the function is undefined at x = -6 and x = 2.

To understand the behavior of G(x) as x approaches -6, simplify the function by canceling the common factor (x + 6) in the numerator and denominator, resulting in G(x) = 1/(x - 2) for x ≠ -6.

Use a graphing calculator to plot the simplified function G(x) = 1/(x - 2). Use the Zoom and Trace features to observe the behavior of the graph as x approaches -6 from both the left and right.

Estimate the y-values on the graph as x approaches -6. Notice the trend in the y-values to support your conclusion about the limit of G(x) as x approaches -6.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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. Understanding limits is crucial for analyzing the behavior of functions near specific values, especially when direct substitution may lead to indeterminate forms. In this context, we are interested in the limit of G(x) as x approaches -6.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visualize the behavior of the function. This visual representation helps in understanding how the function behaves as the input changes, particularly near critical points like asymptotes or discontinuities. In this exercise, using a graphing calculator allows for a more precise estimation of y-values as x approaches -6.

Zoom and Trace Features

The Zoom and Trace features on a graphing calculator enable users to closely examine specific sections of a graph. Zoom allows for adjusting the viewing window to focus on particular intervals, while Trace lets users move along the graph to read y-values corresponding to x-values. These tools are essential for estimating limits and understanding the function's behavior near the point of interest.

Watch next

Master Finding Limits Numerically and Graphically with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Infinite Limits

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

lim x→−5⁻ (3x) / (2x + 10)

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 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

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.

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap

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.

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 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.