Textbook Question
Limits of quotients
Find the limits in Exercises 23–42.
limx→−1 (√(x² + 8) − 3) / (x + 1)
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1. Limits and Continuity
Finding Limits Algebraically
1:38 minutesProblem 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 guidance1
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:
1mKey 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.
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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.
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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.
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