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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1:56 minutes

Problem 55b

Textbook Question

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

lim (x²/2 − 1/x) as

b. x→0⁻

Verified step by step guidance

Step 1: Understand the problem. We need to find the limit of the function \( \frac{x^2}{2} - \frac{1}{x} \) as \( x \) approaches 0 from the left (denoted as \( x \to 0^- \)). This means we are considering values of \( x \) that are slightly less than 0.

Step 2: Analyze the behavior of each term in the function as \( x \to 0^- \). The term \( \frac{x^2}{2} \) approaches 0 because \( x^2 \) becomes very small as \( x \) approaches 0, and dividing by 2 does not affect the limit.

Step 3: Consider the term \( \frac{1}{x} \). As \( x \to 0^- \), \( x \) is negative and very close to 0, so \( \frac{1}{x} \) becomes very large negatively (approaches \( -\infty \)).

Step 4: Combine the behavior of both terms. The term \( \frac{x^2}{2} \) approaches 0, while \( \frac{1}{x} \) approaches \( -\infty \). Therefore, the expression \( \frac{x^2}{2} - \frac{1}{x} \) will be dominated by the \( -\frac{1}{x} \) term, which approaches \( -\infty \).

Step 5: Conclude the limit. Since the dominant term \( -\frac{1}{x} \) approaches \( -\infty \), the limit of the entire expression \( \frac{x^2}{2} - \frac{1}{x} \) as \( x \to 0^- \) is \( -\infty \).

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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 concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit of the function as x approaches 0 from the left (denoted as x→0⁻).

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In the given limit problem, as x approaches 0, the term 1/x becomes significant, potentially leading to an indeterminate form that requires further analysis to resolve. Understanding how to manipulate these forms is essential for finding limits.