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 $\epsilon >0$, there exists a $\delta >0$ such that if $0<|x-0|<\delta$, then $|x^2-0|<\epsilon$.
2. Simplify the expression $|x^2-0|$ to $|x^2|$, which is equal to $|x{|}^2$.
3. To ensure $|x{|}^2<\epsilon$, we need $|x|<\sqrt{\epsilon }$.
4. Choose $\delta =\sqrt{\epsilon }$. This choice of $\delta$ ensures that whenever $0<|x|<\delta$, it follows that $|x{|}^2<\epsilon$.
5. Conclude that the relationship $\delta =\sqrt{\epsilon }$ guarantees the limit $\underset{x\to 0}{lim}{x}^2=0$ 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.