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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2:10 minutes

Problem 57d

Textbook Question

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

lim (x² − 3x + 2) / (x³ − 2x²) as

d. x→2

Verified step by step guidance

First, identify the type of limit problem. This is a rational function limit as x approaches a specific value, x = 2.

Check if direct substitution of x = 2 into the function results in an indeterminate form. Substitute x = 2 into the numerator and denominator: numerator becomes 2² - 3(2) + 2 and denominator becomes 2³ - 2(2²).

Calculate the values from the substitution: numerator becomes 4 - 6 + 2 = 0 and denominator becomes 8 - 8 = 0. This results in the indeterminate form 0/0.

To resolve the indeterminate form, factor both the numerator and the denominator. The numerator x² - 3x + 2 factors into (x - 1)(x - 2) and the denominator x³ - 2x² factors into x²(x - 2).

Cancel the common factor (x - 2) from the numerator and denominator, resulting in the simplified expression (x - 1)/x². Now, substitute x = 2 into the simplified expression to find the limit.

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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 the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given limit problem, both the numerator and denominator are polynomials, which can lead to specific behaviors as x approaches certain values. Understanding how to simplify rational functions is essential for evaluating limits, especially when direct substitution results in indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is crucial, as they often require further analysis, such as factoring, simplifying, or applying L'Hôpital's Rule to resolve the limit correctly.