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

Previous problemNext problem

1:38 minutes

Problem 2.2.55b

Textbook Question

Suppose limx→b f(x) = 7 and lim x→b g(x) = −3. Find

b. limx→b f(x)⋅g(x)

Verified step by step guidance

First, understand the concept of limits. A limit describes the value that a function approaches as the input approaches a certain point. In this problem, we are given the limits of two functions, f(x) and g(x), as x approaches b.

Next, recall the limit multiplication rule: if lim(x→b) f(x) = L and lim(x→b) g(x) = M, then lim(x→b) [f(x)⋅g(x)] = L⋅M. This rule allows us to multiply the limits of two functions directly.

Apply the limit multiplication rule to the given limits: lim(x→b) f(x) = 7 and lim(x→b) g(x) = -3. According to the rule, the limit of the product f(x)⋅g(x) as x approaches b is the product of the individual limits.

Calculate the product of the limits: multiply the limit of f(x), which is 7, by the limit of g(x), which is -3. This gives you the limit of the product f(x)⋅g(x) as x approaches b.

Finally, conclude that the limit of the product f(x)⋅g(x) as x approaches b is the result of the multiplication you performed in the previous step.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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→b f(x) = 7 indicates that as x gets closer to b, the function f(x) approaches the value 7. 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 is the product of their limits. Specifically, if limx→b f(x) = L and limx→b g(x) = M, then limx→b (f(x)⋅g(x)) = L⋅M. This property is essential for solving problems involving the multiplication of functions at a limit.

Evaluating Limits

Evaluating limits involves substituting values or applying limit laws to find the limit of a function as it approaches a specific point. In this scenario, to find limx→b f(x)⋅g(x), one would substitute the known limits of f(x) and g(x) into the product of limits formula, resulting in 7⋅(−3) = −21. This process is crucial for solving limit problems in calculus.