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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2:41 minutes

Problem 56b

Textbook Question

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 1) / (2x + 4) as

b. x→−2⁻

Verified step by step guidance

Identify the type of limit: This is a one-sided limit as x approaches -2 from the left (x → -2⁻).

Substitute x = -2 into the expression (x² − 1) / (2x + 4) to check if it results in an indeterminate form. Calculate the numerator: (-2)² - 1 = 4 - 1 = 3. Calculate the denominator: 2(-2) + 4 = -4 + 4 = 0. The expression is of the form 3/0, indicating a potential vertical asymptote.

Determine the sign of the expression as x approaches -2 from the left. Choose a test value slightly less than -2, such as x = -2.1. Calculate the denominator: 2(-2.1) + 4 = -4.2 + 4 = -0.2, which is negative.

Since the numerator is positive (3) and the denominator is negative as x approaches -2 from the left, the overall expression approaches negative infinity.

Conclude that the limit of (x² − 1) / (2x + 4) as x approaches -2 from the left is -∞.

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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 specific points, including points of discontinuity or infinity. Evaluating limits is essential for determining the continuity of functions and for finding derivatives.

Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomials. Understanding their limits often involves analyzing the behavior of the numerator and denominator as the variable approaches a certain value. In this case, identifying how the function behaves as x approaches -2 is key to determining the limit.