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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2:11 minutes

Problem 2.32

Textbook Question

Evaluate each limit and justify your answer.
lim x→2 (3 / 2x^5−4x^2−50)^4

Verified step by step guidance

Identify the type of limit problem: This is a limit of a function as x approaches a specific value.

Substitute x = 2 into the function to check if it results in an indeterminate form or a defined value.

Calculate the expression inside the parentheses: $2x^5 - 4x^2 - 50$ by substituting x = 2.

Evaluate the expression $3 / (2x^5 - 4x^2 - 50)$ using the result from the previous step.

Raise the result to the power of 4 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

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 evaluating expressions that may not be directly computable at those points.

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. Understanding continuity is essential for evaluating limits, as discontinuities can lead to undefined expressions or different limit values.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers and are combined using addition, subtraction, and multiplication. In the given limit, the polynomial in the denominator can affect the limit's value, especially if it approaches zero, leading to potential indeterminate forms.