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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1:53 minutes

Problem 2.2.17

Textbook Question

Calculating Limits

Find the limits in Exercises 11–22.

limx→−1/2 4x(3x+4)²

Verified step by step guidance

Identify the limit expression: \( \lim_{x \to -\frac{1}{2}} 4x(3x+4)^2 \).

Substitute \( x = -\frac{1}{2} \) directly into the expression to check if it results in a determinate form.

Calculate \( 3x + 4 \) by substituting \( x = -\frac{1}{2} \), which gives \( 3(-\frac{1}{2}) + 4 = -\frac{3}{2} + 4 = \frac{5}{2} \).

Substitute \( x = -\frac{1}{2} \) into \( 4x \), which gives \( 4(-\frac{1}{2}) = -2 \).

Combine the results: \( -2 \times (\frac{5}{2})^2 \) to find the limit. Simplify the expression to complete the calculation.

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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, which is crucial for defining derivatives and integrals. In this case, we are interested in the limit of the function as x approaches -1/2.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function in the limit problem, 4x(3x+4)², is a polynomial function, and understanding its structure is essential for evaluating limits. Polynomial functions are continuous everywhere, which simplifies the process of finding limits.

Substitution Method

The substitution method is a technique used to evaluate limits by directly substituting the value that x approaches into the function, provided the function is continuous at that point. If direct substitution results in an indeterminate form, further algebraic manipulation may be necessary. In this case, substituting x = -1/2 into the polynomial will help find the limit.