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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1:43 minutes

Problem 31

Textbook Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→2 (5x−6)^3/2

Verified step by step guidance

Identify the type of limit problem: This is a direct substitution problem where we need to find the limit of a function as x approaches a specific value.

Recognize the function: The function given is \((5x - 6)^{3/2}\).

Apply direct substitution: Substitute \(x = 2\) into the function to evaluate the limit.

Calculate the expression inside the function: Compute \(5(2) - 6\) to simplify the expression inside the power.

Evaluate the power: Once the expression inside the function is simplified, raise it to the power of \(\frac{3}{2}\) 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 continuity and differentiability. In this case, we are interested in the limit of the function as x approaches 2.

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. For the limit to exist, the function must not have any breaks, jumps, or asymptotes at the point of interest. The expression (5x−6) is a polynomial, which is continuous everywhere, including at x = 2.

Evaluating Limits of Composite Functions

When evaluating limits of composite functions, such as (5x−6) raised to the power of 3/2, it is essential to first find the limit of the inner function before applying the outer function. This process often involves substituting the limit value into the inner function and then applying the outer function to that result, ensuring that the operations are valid within the domain of the functions involved.

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