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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2:19 minutes

Problem 21

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→−95x

Verified step by step guidance

Identify the type of limit: This is a limit of a linear function as \( x \) approaches a constant.

Recognize that the function \( 5x \) is continuous everywhere, including at \( x = -95 \).

Apply the property of limits for continuous functions: \( \lim_{x \to a} f(x) = f(a) \).

Substitute \( x = -95 \) into the function \( 5x \) to evaluate the limit.

Conclude that the limit exists and is equal to the value of the function at \( x = -95 \).

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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 -9.

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. Continuous functions do not have breaks, jumps, or holes, making it easier to evaluate limits. The function 5x is continuous everywhere, which simplifies the process of finding its limit.

Substitution in Limits

Substitution is a technique used in limit evaluation where you directly replace the variable in the function with the value it approaches. If the function is continuous at that point, this method yields the limit's value. For the limit lim x→−95x, substituting -9 directly into the function will provide the limit's value.