Textbook Question
Limits of quotients
Find the limits in Exercises 23–42.
limx→−3 (2 − √(x² − 5)) / (x + 3)
1. Limits and Continuity
Finding Limits Algebraically
1:50 minutesProblem 2.2.56a
Textbook Question
Suppose that limx→−2 p(x) = 4, limx→−2 r(x) = 0, and limx→−2 s(x) = −3. Find
a. limx→−2 (p(x) + r(x) + s(x))
Verified step by step guidance1
Step 1: Understand the problem statement. We are given the limits of three functions as x approaches -2: lim(x→−2) p(x) = 4, lim(x→−2) r(x) = 0, and lim(x→−2) s(x) = −3. We need to find the limit of the sum of these functions as x approaches -2.
Step 2: Recall the property of limits that states the limit of a sum is the sum of the limits. This means that lim(x→−2) (p(x) + r(x) + s(x)) = lim(x→−2) p(x) + lim(x→−2) r(x) + lim(x→−2) s(x).
Step 3: Substitute the given limits into the equation from Step 2. We have lim(x→−2) p(x) = 4, lim(x→−2) r(x) = 0, and lim(x→−2) s(x) = −3.
Step 4: Add the limits together: 4 + 0 + (-3).
Step 5: The result of the addition gives us the limit of the sum of the functions as x approaches -2.
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.
Limits of Functions
A limit describes the value that a function approaches as the input approaches a certain point. In this case, we are interested in the behavior of the functions p(x), r(x), and s(x) as x approaches -2. Understanding limits is crucial for evaluating expressions involving functions at points where they may not be explicitly defined.
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Limit Laws
Limit laws are rules that allow us to compute the limit of a sum, difference, product, or quotient of functions based on the limits of the individual functions. For example, the limit of a sum is the sum of the limits, provided the limits exist. This principle will be applied to find the limit of the expression p(x) + r(x) + s(x) as x approaches -2.
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Evaluating Limits
Evaluating limits involves substituting the point of interest into the function, if possible, or applying limit laws to simplify the expression. In this problem, we will substitute the limits of p(x), r(x), and s(x) as x approaches -2 to find the overall limit of their sum. This process is essential for solving the given limit problem.
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