Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→9 √(x − 5) = 2

검증된 단계별 안내

Step 1: Understand the formal definition of a limit. The limit of a function f(x) as x approaches a value c is L if for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε.

Step 2: Identify the function and the limit statement. Here, the function is f(x) = √(x - 5) and we want to prove that lim(x→9) f(x) = 2.

Step 3: Set up the inequality |√(x - 5) - 2| < ε. We need to find a δ such that whenever 0 < |x - 9| < δ, this inequality holds.

Step 4: Manipulate the inequality |√(x - 5) - 2| < ε to find a suitable δ. Start by squaring both sides to eliminate the square root, leading to |x - 5 - 4| < ε².

Step 5: Solve the inequality |x - 9| < δ by relating it to the previous inequality. Choose δ such that it satisfies both the original limit condition and the manipulated inequality, ensuring the limit holds as x approaches 9.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Limit Definition

The formal definition of a limit, often called the epsilon-delta definition, states that for a function f(x) to have a limit L as x approaches a value c, for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This definition is crucial for rigorously proving limit statements.

Square Root Function

Understanding the behavior of the square root function is essential, as it is continuous and differentiable for x > 0. In this context, the function √(x - 5) is involved, and knowing its properties helps in manipulating and evaluating the limit as x approaches a specific value, ensuring the function behaves predictably.

Substitution Method

The substitution method is a technique used to simplify limit problems by introducing a new variable. For the given limit, setting u = x - 5 transforms the problem into a simpler form, allowing us to apply the epsilon-delta definition more easily. This method helps in isolating the variable and focusing on the core behavior of the function near the limit point.

Using the Formal DefinitionProve the limit statements in Exercises 37–50.limx→9 √(x − 5) = 2