Problem 27 - Calculus: Early Transcendentals

Question

Prove the statement using the \( \varepsilon \), \( \delta \) definition of a limit. \( \displaystyle \lim_{x \to 0} | x | = 0 \)

Step-by-Step Solution

Answer

Set \(\delta = \varepsilon\). If \(0 < |x| < \delta\), then \(| x | < \varepsilon\).

Step 1: Understanding the Limit Definition

To prove that \( \lim_{x \to 0} |x| = 0 \) using the \( \varepsilon \), \( \delta \) definition, we need to show that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x| < \delta \), then \(| |x| - 0 | < \varepsilon\).

Step 2: Express the Problem

The inequality we need to satisfy is \(|x| < \varepsilon\) if \(0 < |x| < \delta\). The absolute value \(|x|\) is already positive, so \(|x|\) itself is the expression we need to bound by \(\varepsilon\).

Step 3: Determine Delta

Since the inequality \(|x| < \varepsilon\) is needed for \(0 < |x| < \delta\), we can choose \(\delta = \varepsilon\). This choice makes sure that whenever \(0 < |x| < \delta\), it follows directly that \(|x| < \varepsilon\).

Step 4: Verification

Let's verify this choice. Suppose \(0 < |x| < \delta = \varepsilon\). Then naturally \(|x| < \varepsilon\) is satisfied, proving the statement according to the \(\varepsilon\), \(\delta\) definition of limits.

Key Concepts

calculus

Calculus is a branch of mathematics that deals with continuous change. This field is crucial for understanding concepts like derivatives, integrals, and limits.
In this context, limits play a fundamental role. They help us describe the behavior of functions as they approach certain points.
Imagine trying to find out what happens to the function as the input gets closer to a particular value. This is exactly what limits are used for.

• If you've ever calculated the slope of a curve or the area under it, you've used calculus.
• It's all about understanding how functions behave when close to certain points.

Calculus also allows us to tackle problems in physics, engineering, economics, and more. For instance, finding the instantaneous rate of change of a moving car or determining the total accumulated amount of rain over a day.
By using limits, calculus enables these kinds of predictions and calculations.

limit of absolute value

A limit of an absolute value function is a specific type of limit in calculus. Here, we deal with the behavior of absolute value expressions as they approach a particular point.
For example, considering the limit as the input approaches zero, such as the exercise given with \(\lim_{x \to 0} |x| = 0\).
Absolute values represent the distance from zero without regard to direction.
This means values can never be negative. Understanding how absolute value affects limits is crucial for precisely evaluating functions.

• Absolute value ensures the limit focuses on magnitude rather than sign.
• It helps in scenarios involving positive distances regardless of original sign.

In the exercise, realizing that the limit evaluates to zero as x approaches zero simplifies how we approach proving it.

epsilon-delta proof

An epsilon-delta proof is a mathematical way of proving the limit of a function as it approaches a point.
This approach focuses on formalizing how close function values get to an expected limit.
In simpler terms, it establishes a rule for making function outputs as close as we want to a limit by choosing an input sufficiently close.

• \(\varepsilon\) (epsilon) represents a small positive distance from the limit we can tolerate.
• \(\delta\) (delta) indicates a small positive range around the input value approaching zero.

Suppose we want to prove \(\lim_{x \to 0} |x| = 0\) using epsilon-delta proof. We argue that for every \(\varepsilon > 0\), there is a \(\delta > 0\) making sure whenever \(0 < |x| < \delta\), it implies \(| |x| - 0 | < \varepsilon\).
By picking \(\delta = \varepsilon\), we ensure this occurs, validating the limit statement. Hence, the epsilon-delta method confirms that the limits are both mathematically sound and precise.