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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1:46 minutes

Problem 2.29

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→3 −5x / √4x − 3

Verified step by step guidance

Identify the limit expression: \( \lim_{{x \to 3}} \frac{{-5x}}{{\sqrt{{4x}} - 3}} \).

Substitute \( x = 3 \) into the expression to check if it results in an indeterminate form.

Calculate \( \sqrt{{4 \times 3}} - 3 \) to see if the denominator becomes zero.

If the expression is indeterminate, consider rationalizing the denominator or using L'Hôpital's Rule if applicable.

Evaluate the limit using the chosen method to simplify the expression.

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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 3.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit different behaviors depending on the values of x, particularly at points where the denominator is zero. Understanding how to simplify and analyze these functions is essential for finding limits, especially when approaching points that may lead to indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is crucial, as they often require additional techniques, such as L'Hôpital's Rule or algebraic manipulation, to resolve and find the actual limit.

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