Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→1 1/x = 1

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Understand the formal definition of a limit: For a function f(x) and a limit L, the statement lim x→c f(x) = L means that for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε.

Identify the components of the limit statement: Here, f(x) = 1/x, L = 1, and c = 1. We need to show that for every ε > 0, there exists a δ > 0 such that if 0 < |x - 1| < δ, then |1/x - 1| < ε.

Express the condition |1/x - 1| < ε in terms of x: Start by simplifying the expression |1/x - 1|. This can be rewritten as |(1 - x)/x|. We want this to be less than ε.

Find a suitable δ: To ensure |(1 - x)/x| < ε, we can manipulate the inequality. Notice that |(1 - x)/x| = |1 - x|/|x|. We need to find δ such that |1 - x| < ε|x|. Assume |x - 1| < 1, which implies 0 < x < 2, and thus 1/2 < 1/x < 1. This helps in bounding |x|.

Conclude the proof: Choose δ = min(1, ε/2). This choice ensures that if 0 < |x - 1| < δ, then |1/x - 1| < ε, satisfying the formal definition of the limit. Therefore, lim x→1 1/x = 1 is proven using the formal definition.

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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) to have 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 selecting an appropriate δ for a given ε to show that the function's output can be made arbitrarily close to the limit by choosing inputs sufficiently close to the point of interest.

Continuity and Limits

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. Understanding continuity helps in analyzing limits, as continuous functions behave predictably near their points, making it easier to apply the epsilon-delta definition.

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