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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1:21 minutes

Problem 55d

Textbook Question

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

lim (x²/2 − 1/x) as

d. x→−1

Verified step by step guidance

Identify the function for which you need to find the limit: \( f(x) = \frac{x^2}{2} - \frac{1}{x} \).

Substitute \( x = -1 \) into the function to evaluate the limit directly: \( f(-1) = \frac{(-1)^2}{2} - \frac{1}{-1} \).

Calculate each term separately: \( \frac{(-1)^2}{2} = \frac{1}{2} \) and \( -\frac{1}{-1} = 1 \).

Combine the results from the previous step: \( \frac{1}{2} + 1 \).

Simplify the expression to find the limit as \( x \to -1 \).

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

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. Understanding rational functions is crucial for limit evaluation, especially when determining behavior as the variable approaches a specific value. In this case, the function involves a polynomial in the numerator and a term involving x in the denominator, which can affect the limit as x approaches -1.

Substitution in Limits

Substitution is a technique used in limit evaluation where you directly replace the variable with the value it approaches. This method is effective when the function is continuous at that point. However, if direct substitution leads to an indeterminate form, further analysis or algebraic manipulation may be necessary to find the limit.