Using the Formal Definition

Each of Exercises 31–36 gives a function f(x), a point c, and a positive number ε. Find L = lim x→c f(x). Then find a number δ > 0 such that |f(x)−L| < ε whenever 0 < |x−c| < δ.

f(x) = −3x − 2, c = −1, ε = 0.03

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Step 1: Begin by finding the limit L = lim x→c f(x). Substitute c = -1 into the function f(x) = -3x - 2 to find L. This involves calculating f(-1).

Step 2: Calculate f(-1) by substituting x = -1 into the function: f(-1) = -3(-1) - 2. Simplify this expression to find the value of L.

Step 3: With L determined, we need to find δ > 0 such that |f(x) - L| < ε whenever 0 < |x - c| < δ. Start by expressing |f(x) - L| in terms of x.

Step 4: Substitute f(x) = -3x - 2 and L into the inequality |f(x) - L| < ε. This gives |(-3x - 2) - L| < 0.03. Simplify this expression to find a condition on x.

Step 5: Solve the inequality |(-3x - 2) - L| < 0.03 for x, which will involve isolating |x + 1|. Determine δ such that 0 < |x + 1| < δ satisfies the inequality.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limit of a Function

The limit of a function describes the value that the function approaches as the input approaches a certain point. In this case, we need to find L = lim x→c f(x), which means evaluating the function f(x) at the point c and determining the value it approaches as x gets closer to c.

Epsilon-Delta Definition of a Limit

The epsilon-delta definition formalizes the concept of limits. It states that for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This means we can make the function's output as close to L as desired by restricting x sufficiently close to c.

Finding δ for Given ε

To find δ for a given ε, we analyze the inequality |f(x) - L| < ε. This involves manipulating the expression to isolate |x - c|, allowing us to determine a suitable δ that ensures the function's output remains within the ε range of L when x is within δ of c.

Using the Formal DefinitionEach of Exercises 31–36 gives a function f(x), a point c, and a positive number ε. Find L = lim x→c f(x). Then find a number δ > 0 such that |f(x)−L| < ε whenever 0 < |x−c| < δ.f(x) = −3x − 2, c = −1, ε = 0.03