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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1:48 minutes

Problem 2.31

Textbook Question

Evaluate each limit and justify your answer.
lim x→0 (x^8−3x^6−1)^40

Verified step by step guidance

Identify the limit expression: $\lim_{x \to 0} (x^8 - 3x^6 - 1)^{40}$.

Evaluate the expression inside the limit as $x$ approaches 0: $x^8 - 3x^6 - 1$. Substitute $x = 0$ to get $0^8 - 3(0)^6 - 1 = -1$.

Since the expression inside the limit approaches a constant value, $-1$, as $x$ approaches 0, the limit simplifies to $(-1)^{40}$.

Recognize that $(-1)^{40}$ is a power of $-1$. Since 40 is an even number, $(-1)^{40}$ equals 1.

Conclude that the limit is 1, as the expression simplifies to a constant value.

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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. In this case, we are interested in the limit of the function as x approaches 0. Understanding limits is crucial for evaluating functions that may not be directly computable at specific points.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The expression in the limit, (x^8−3x^6−1), is a polynomial function. Analyzing polynomial functions helps in determining their behavior at specific points, such as identifying leading terms and their contributions to the limit.

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. In evaluating the limit of the given polynomial raised to a power, we can apply the property of continuity, which allows us to substitute the limit value directly into the function, simplifying the evaluation process.