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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2:58 minutes

Problem 57a

Textbook Question

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

lim (x² − 3x + 2) / (x³ − 2x²) as

a. x→0⁺

Verified step by step guidance

Step 1: Begin by substituting x = 0 into the function \((x^2 - 3x + 2) / (x^3 - 2x^2)\) to check if the limit can be directly evaluated. This will help identify if the function is undefined at x = 0.

Step 2: Since direct substitution results in an indeterminate form \(0/0\), apply L'Hôpital's Rule, which is used to evaluate limits of indeterminate forms. Differentiate the numerator \(x^2 - 3x + 2\) and the denominator \(x^3 - 2x^2\) separately.

Step 3: Differentiate the numerator: \(\frac{d}{dx}(x^2 - 3x + 2) = 2x - 3\). Differentiate the denominator: \(\frac{d}{dx}(x^3 - 2x^2) = 3x^2 - 4x\).

Step 4: Substitute x = 0 into the differentiated function \((2x - 3) / (3x^2 - 4x)\) to evaluate the limit as x approaches 0 from the positive side.

Step 5: Analyze the behavior of the function as x approaches 0 from the positive side. Consider the signs of the numerator and denominator to determine if the limit approaches \(\infty\), \(-\infty\), or a finite value.

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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

Rational functions are expressions formed by the ratio of two polynomials. In the given limit problem, the function consists of a polynomial in the numerator and another in the denominator. Understanding how to simplify and analyze rational functions is essential for finding limits, especially when dealing with indeterminate forms.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side, either the left (x→c⁻) or the right (x→c⁺). In this problem, evaluating the limit as x approaches 0 from the positive side (0⁺) is important for understanding the function's behavior near that point, particularly when the function may behave differently from each side.