Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.
lim x→−3 |2x|=6 (Hint: Use the inequality ∥a|−|b∥≤|a−b|, which holds for all constants a and b (see Exercise 74).)

Verified step by step guidance

1. Step 1: Start by recalling the precise definition of a limit. For the limit \( \lim_{x \to -3} |2x| = 6 \) to hold, for every \( \varepsilon > 0 \), there must exist a \( \delta > 0 \) such that if \( 0 < |x + 3| < \delta \), then \( ||2x| - 6| < \varepsilon \).
2. Step 2: Express the function in terms of the limit. We have \( |2x| = 2|x| \). Therefore, we need to show that \( |2|x| - 6| < \varepsilon \) whenever \( 0 < |x + 3| < \delta \).
3. Step 3: Simplify the inequality \( |2|x| - 6| < \varepsilon \). This can be rewritten as \( |2x - 6| < \varepsilon \).
4. Step 4: Relate \( |2x - 6| \) to \( |x + 3| \). Notice that \( |2x - 6| = 2|x + 3| \). Therefore, the inequality \( |2x - 6| < \varepsilon \) becomes \( 2|x + 3| < \varepsilon \).
5. Step 5: Solve for \( \delta \) in terms of \( \varepsilon \). From \( 2|x + 3| < \varepsilon \), we get \( |x + 3| < \frac{\varepsilon}{2} \). Thus, choose \( \delta = \frac{\varepsilon}{2} \) to satisfy the condition.

Key Concepts

Limit Definition

The precise definition of a limit states that for a function f(x) to approach a limit L as x approaches a value c, for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This formalizes the intuitive idea of limits and is essential for proving limit statements rigorously.

Absolute Value and Inequalities

The absolute value function measures the distance of a number from zero on the number line. The inequality ∥a|−|b∥≤|a−b| helps in establishing relationships between the absolute values of expressions, which is crucial when manipulating limits involving absolute values, such as |2x| in this case.

Epsilon-Delta Relationship

In limit proofs, the relationship between ε and δ is critical. It specifies how close x must be to c (within δ) to ensure that f(x) is within ε of L. Establishing this relationship is key to demonstrating that the limit exists and is valid under the conditions set by the definition of a limit.