Problem 25 - Calculus Early Transcendentals

Question

Prove the statement using the \(\varepsilon_{t} \delta\) definition of limit. \(\lim _{x \rightarrow 0} x^{2}=0\)

Step-by-Step Solution

Answer

The limit is proven using \( \delta = \sqrt{\varepsilon} \) to satisfy \( |x^2 - 0| < \varepsilon \).

Step 1: Understanding the Problem

We need to prove that the function \( f(x) = x^2 \) has a limit of 0 as \( x \) approaches 0, using the \( \varepsilon \)–\( \delta \) definition of a limit.

Step 2: Recall the Definition of a Limit

For \( \lim_{x \to a} f(x) = L \) to hold, for every \( \varepsilon > 0 \), there must exist a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \varepsilon \). In this case, our \( a = 0 \) and \( L = 0 \).

Step 3: Set up the Inequality |f(x) - L| < ε

We start with \( |x^2 - 0| < \varepsilon \), which simplifies to \( |x^2| < \varepsilon \). Since \( x^2 \) is always non-negative, this becomes \( x^2 < \varepsilon \).

Step 4: Relate x with δ

We need \( |x| < \delta \) to ensure \( x^2 < \varepsilon \). Observe that if \( |x| < \delta \), then \( x^2 < \delta^2 \). We want \( \delta^2 < \varepsilon \), so we choose \( \delta = \sqrt{\varepsilon} \).

Step 5: Prove the Choice of δ

Suppose we choose \( \delta = \sqrt{\varepsilon} \). If \( 0 < |x| < \delta \), then \( x^2 < \delta^2 = \varepsilon \). Hence, \( |x^2 - 0| = |x^2| < \varepsilon \), which satisfies our limit condition.

Step 6: Conclusion

Using \( \delta = \sqrt{\varepsilon} \), we have shown that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) ensuring \( |x^2| < \varepsilon \) whenever \( 0 < |x| < \delta \). Thus, \( \lim_{x \to 0} x^2 = 0 \) is proven.

Key Concepts

Epsilon-Delta Definition

When we talk about the limit of a function using the Epsilon-Delta (\(\varepsilon\)-\(\delta\)) definition, we step into a fundamental concept in calculus that rigorously describes how functions behave as their inputs approach certain values. This definition aims to capture precisely what we mean when we say a function approaches a particular value.The core of the \(\varepsilon\)-\(\delta\) definition of a limit is:

• For every error margin \(\varepsilon > 0\) (no matter how tiny you choose it), there exists a proximity range \(\delta > 0\) such that for all \(x\) within this range, the function's outputs are within the error margin of the limit.
• In practical terms, if we claim that \(\lim_{x \to a} f(x) = L\), then whenever \(0 < |x-a| < \delta\), the function satisfies \(|f(x) - L| < \varepsilon\). This holds even as \(x\) creeps infinitely close to \(a\).

This definition is crucial in mathematics as it forms the basis for proving that limits exist precisely and is extensively used to evaluate and understand continuous behavior in functions.

Limit Proof

Proving the limit of a function involves demonstrating, step by step, that the function adheres to the \(\varepsilon\)-\(\delta\) definition. Let's consider a specific example: proving \(\lim_{x \to 0} x^2 = 0\).Here's how we approach the proof:

• Set the expected limit \(L = 0\) and the approaching point \(a = 0\).
• To prove \(x^2\) reaches \(0\) as \(x\) closes in on \(0\), start with the \(\varepsilon\)-condition: we need \(|x^2| < \varepsilon\).
• By observing \(x^2\) is always non-negative, we establish that \(x^2 < \varepsilon\) when \(|x| < \delta\), knowing \(\delta^2\) should equal \(\varepsilon\).
• This leads to selecting \(\delta = \sqrt{\varepsilon}\), ensuring our proof holds for any \(\varepsilon > 0\).

Proofs like these showcase the beauty of calculus by providing a logical path from assumptions to conclusions, grounded in solid mathematical reasoning.

Continuous Functions

Continuous functions are an essential concept in calculus, representing functions that have no sudden jumps, breaks, or gaps in their graphs. Understanding them is key to interpreting the behaviors and trends in various real-world scenarios.A function \(f(x)\) is said to be continuous at a point \(x = a\) if the following criteria are met:

• The function \(f(x)\) is defined at \(x = a\).
• There exists a limit \(\lim_{x \to a} f(x)\).
• That limit is exactly \(f(a)\), meaning \(\lim_{x \to a} f(x) = f(a)\).

In simple terms, this implies that you can draw the graph of \(f(x)\) without lifting your pencil from \(x = a\).Continuous functions play a significant role in various areas of mathematics and sciences, particularly where smooth changes over intervals are essential, like in physics and engineering.