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

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) $.

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 $.

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 $.

Step 4: Substitute the simplified expression into the limit condition: $ \left| (x + 4) - 8 \right| = |x - 4| $.

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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