Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx →4 (9 − x) = 5

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Understand the formal definition of a limit: For a function f(x) and a limit L as x approaches a value c, we say that 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.

Identify the components of the limit statement: Here, f(x) = 9 - x, L = 5, and c = 4. We need to show that for every ε > 0, there exists a δ > 0 such that if 0 < |x - 4| < δ, then |(9 - x) - 5| < ε.

Simplify the expression |(9 - x) - 5|: This simplifies to |4 - x|. We need to ensure that |4 - x| < ε whenever 0 < |x - 4| < δ.

Choose δ = ε: By setting δ = ε, we ensure that whenever 0 < |x - 4| < δ, it follows that |4 - x| < ε, satisfying the condition for the limit.

Conclude the proof: Since for every ε > 0, we can find δ = ε such that 0 < |x - 4| < δ implies |4 - x| < ε, we have proven that lim(x→4) (9 - x) = 5 using the formal definition of a limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit Definition

The formal definition of a limit states that for a function f(x), the limit as x approaches a value 'a' is L if, for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - a| < δ, it follows that |f(x) - L| < ε. This definition is crucial for rigorously proving limit statements.

Epsilon-Delta Proof

An epsilon-delta proof is a method used to demonstrate the validity of a limit using the formal definition. It involves selecting an appropriate δ for a given ε to show that the function's output can be made arbitrarily close to the limit as the input approaches the specified value.

Function Behavior Near a Point

Understanding how a function behaves near a specific point is essential for limit proofs. In this case, analyzing the function f(x) = 9 - x as x approaches 4 helps to determine if the output approaches the limit of 5, which is necessary for completing the proof.

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