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→4 x^2−16 / x−4=8 (Hint: Factor and simplify.)

Verified step by step guidance

1. Step 1: Recognize that the expression \( \frac{x^2 - 16}{x - 4} \) is undefined at \( x = 4 \). To simplify, factor the numerator: \( x^2 - 16 = (x - 4)(x + 4) \).
2. Step 2: Simplify the expression by canceling the common factor \( x - 4 \) in the numerator and denominator, resulting in \( x + 4 \) for \( x \neq 4 \).
3. Step 3: According to the definition of a limit, for every \( \varepsilon > 0 \), there must exist a \( \delta > 0 \) such that if \( 0 < |x - 4| < \delta \), then \( \left| \frac{x^2 - 16}{x - 4} - 8 \right| < \varepsilon \).
4. Step 4: Substitute the simplified expression into the limit condition: \( \left| (x + 4) - 8 \right| = |x - 4| \).
5. Step 5: To satisfy the limit condition, choose \( \delta = \varepsilon \). This ensures that whenever \( 0 < |x - 4| < \delta \), \( |x - 4| < \varepsilon \), proving the limit exists and equals 8.

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.

Factoring and Simplifying

Factoring and simplifying expressions is a key technique in calculus, especially when evaluating limits. In the given limit, the expression x^2 - 16 can be factored as (x - 4)(x + 4), allowing for cancellation of the (x - 4) term in the denominator, which simplifies the limit evaluation.

Epsilon-Delta Relationship

In the context of limits, the relationship between ε and δ is essential for proving that a limit exists. For the limit lim x→4 (x^2 - 16)/(x - 4) = 8, one must find a δ such that when x is within δ of 4, the value of the function is within ε of 8. This relationship ensures that the function behaves as expected near the limit point.