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))

Guida verificata passo dopo passo

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.

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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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Pratica correlata

Domanda del libro di testo

Use the graph of the greatest integer function y = ⌊x⌋, Figure 1.10 in Section 1.1, to help you find the limits in Exercises 21 and 22.

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b. limt→4−(t−⌊t⌋)

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Domanda del libro di testo

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.