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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1:40 minutes

Problem 2.2.53

Textbook Question

Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find

b. limx→c 2f(x)g(x)

Verified step by step guidance

First, understand that the problem involves finding the limit of a product of functions as x approaches a certain value, c.

Recall the limit property for products: if lim(x→c) f(x) = L and lim(x→c) g(x) = M, then lim(x→c) [f(x) * g(x)] = L * M.

In this problem, you are given that lim(x→c) f(x) = 5 and lim(x→c) g(x) = -2.

Apply the limit property for products to find lim(x→c) [2 * f(x) * g(x)]. This can be rewritten as 2 * lim(x→c) [f(x) * g(x)].

Substitute the known limits into the expression: 2 * (5 * -2). Calculate this expression 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.

Limit of a Function

The limit of a function describes the value that a function approaches as the input approaches a certain point. In this case, limx→c f(x) = 5 indicates that as x gets closer to c, the function f(x) approaches the value 5. Understanding limits is fundamental in calculus as it lays the groundwork for concepts like continuity and derivatives.

Product of Limits

The product of limits states that if the limits of two functions exist as x approaches a certain value, then the limit of their product can be found by multiplying the individual limits. Specifically, if limx→c f(x) = L and limx→c g(x) = M, then limx→c (f(x)g(x)) = L * M. This property is essential for solving the given limit problem involving the functions f(x) and g(x).

Constant Multiplication in Limits

When calculating limits, multiplying a function by a constant does not affect the limit's existence. Specifically, if k is a constant and limx→c f(x) = L, then limx→c (k * f(x)) = k * L. This concept is crucial for evaluating the limit limx→c 2f(x)g(x), as the constant 2 can be factored out when calculating the limit.