Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx→3 (3x − 7) = 2

Verified step by step guidance

1. Understand the formal definition of a limit: For a function f(x) and a limit L as x approaches a value c, we say lim(x→c) f(x) = L if for every ε > 0, there exists a δ > 0 such that 0 < |x - c| < δ implies |f(x) - L| < ε.
2. Identify the components of the problem: Here, f(x) = 3x - 7, c = 3, and L = 2. We need to show that for every ε > 0, there exists a δ > 0 such that if 0 < |x - 3| < δ, then |(3x - 7) - 2| < ε.
3. Simplify the expression |(3x - 7) - 2|: This becomes |3x - 9|, which can be further simplified to 3|x - 3|.
4. Relate |3x - 9| to ε: We want 3|x - 3| < ε. This implies |x - 3| < ε/3. Therefore, we can choose δ = ε/3.
5. Conclude the proof: For every ε > 0, if we choose δ = ε/3, then whenever 0 < |x - 3| < δ, it follows that |3x - 9| < ε. Thus, by the formal definition of a limit, lim(x→3) (3x - 7) = 2.

Key Concepts

Limit Definition

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

Epsilon-Delta Proof

An epsilon-delta proof is a method used to demonstrate the validity of a limit using the formal definition. It involves finding a suitable δ for a given ε, ensuring that the function's output remains within ε of the limit L when the input is within δ of the point a. This technique is essential for establishing the precise behavior of functions near specific points.

Function Behavior Near a Point

Understanding how a function behaves near a specific point is vital for limit proofs. In this case, analyzing the function f(x) = 3x - 7 as x approaches 3 helps determine if it indeed approaches the limit of 2. This involves substituting values close to 3 into the function and observing the output, which reinforces the concept of continuity and limit existence.