Problem 23 - Calculus: Early Transcendentals

Question

Prove the statement using the \( \varepsilon \), \( \delta \) definition of a limit. \( \displaystyle \lim_{x \to a} x = a \)

Step-by-Step Solution

Answer

The limit \( \lim_{x \to a} x = a \) holds because choosing \( \delta = \varepsilon \) satisfies the conditions for any \( \varepsilon > 0 \).

Step 1: Understanding the Limit Definition

The limit statement \( \lim_{{x \to a}} x = a \) means that as \( x \) approaches \( a \), the value of \( x \) gets arbitrarily close to \( a \). To prove this using the \( \varepsilon \) and \( \delta \) definition of a limit, for every \( \varepsilon > 0 \), we need to find a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |x - a| < \varepsilon \).

Step 2: Expression of the Condition \(|x - a| < \varepsilon\)

From the limit definition, our task translates to ensuring that \(|x - a|\) (distance between \(x\) and \(a\)) remains less than \(\varepsilon\). Our goal is to prove \(|x - a| < \varepsilon\) given \(0 < |x - a| < \delta\).

Step 3: Choosing \( \delta \)

To satisfy \( |x - a| < \varepsilon \), we can straightforwardly choose \( \delta = \varepsilon \). This choice works because as long as \( |x - a| < \varepsilon \), both conditions \( |x - a| < \varepsilon \) and \( |x-a| < \delta \) are satisfied by definition since we set \( \delta = \varepsilon \).

Step 4: Conclusion of Proof

With \( \delta = \varepsilon \), the conditions \( 0 < |x - a| < \delta \) naturally imply \( |x - a| < \varepsilon \). Hence, the \( \varepsilon \), \( \delta \) definition for the limit \( \lim_{{x \to a}} x = a \) holds true for any given \( \varepsilon > 0 \). This completes the proof.

Key Concepts

Limit of a Function

The concept of a limit is central to calculus and mathematical analysis. The limit of a function describes how a function behaves as the input (usually denoted as \( x \)) approaches a particular value (denoted \( a \)). In simple terms, you can think of a limit as the value that a function approaches, but does not necessarily reach, as the input gets closer and closer to the specified number.

When we write \( \lim_{{x \to a}} f(x) = L \), it means as \( x \) gets closer to \( a \), the value of the function \( f(x) \) approaches \( L \). The limit doesn't always have to be the function's value at \( a \).

• We focus on behavior very close to \( a \), not necessarily exactly at \( a \).
• If you can make \( f(x) \) as close as you want to \( L \) just by choosing \( x \) sufficiently near \( a \), then \( L \) is the limit of \( f \) as \( x \) approaches \( a \).

Understanding limits is essential for grasping other concepts in calculus, like derivatives and integrals.

Proof Techniques

Proofs come in many forms and serve as a way to rigorously demonstrate the truth of mathematical statements. One common method is the epsilon-delta proof, which is particularly used for proving limits in calculus.

In epsilon-delta proofs, we need to show that for every \( \varepsilon > 0 \) (no matter how small), there exists a \( \delta > 0 \) such that if **\( 0 < |x - a| < \delta \)**, then **\( |f(x) - L| < \varepsilon \)**. This statement ensures that \( f(x) \) can get arbitrarily close to \( L \) as \( x \) approaches \( a \).

• Identify your \( \varepsilon \): the level of closeness to the limit.
• Determine your \( \delta \): the acceptable range for the input \( x \).
• Show that if \( x \) is within \( \delta \) of \( a \), then \( f(x) \) is within \( \varepsilon \) of \( L \).

This proof technique gives a rigorous approach to establishing limits and demonstrating continuity.

Continuity

Continuity is a property of functions that describes smooth and unbroken behavior over an interval. Intuitively, if you can draw a function without lifting your pencil from the paper, it is continuous. In mathematical terms, a function \( f(x) \) is continuous at a point \( a \) if three conditions are satisfied:

• The function \( f(x) \) is defined at \( a \).
• The limit \( \lim_{{x \to a}} f(x) \) exists.
• The limit is equal to the function's value at \( a \), that is, \( \lim_{{x \to a}} f(x) = f(a) \).

A continuous function has no jumps, holes, or asymptotes at the point \( a \).

Understanding continuity is important as it underpins various operations in calculus, like differentiation and integration. Continuous functions guarantee smoother transitions and are fundamental in defining problem setups in calculus applications.