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→0 x^2=0 (Hint: Use the identity √x2=|x|.)

Verified step by step guidance

1. Start by recalling the precise definition of a limit: For every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - 0| < \delta \), then \( |x^2 - 0| < \varepsilon \).
2. Simplify the expression \( |x^2 - 0| \) to \( |x^2| \), which is equal to \( |x|^2 \).
3. To ensure \( |x|^2 < \varepsilon \), we need \( |x| < \sqrt{\varepsilon} \).
4. Choose \( \delta = \sqrt{\varepsilon} \). This choice of \( \delta \) ensures that whenever \( 0 < |x| < \delta \), it follows that \( |x|^2 < \varepsilon \).
5. Conclude that the relationship \( \delta = \sqrt{\varepsilon} \) guarantees the limit \( \lim_{x \to 0} x^2 = 0 \) exists.

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 a, for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. This formalism is crucial for proving limits rigorously.

Epsilon-Delta Relationship

In the context of limits, the ε (epsilon) represents how close f(x) must be to the limit L, while δ (delta) represents how close x must be to the point a. Establishing a relationship between ε and δ is essential to demonstrate that as x gets sufficiently close to a, f(x) will be within ε of L.

Absolute Value and Squaring

The identity √x² = |x| is important in this limit proof because it allows us to express x² in terms of its absolute value. This is particularly useful when dealing with limits approaching zero, as it simplifies the analysis of the function's behavior near that point.