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→7 f(x)=9, where f(x)={3x−12 if x≤7
x+2 if x>7

Key Concepts

Definition of a Limit

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 must exist a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This formal definition is crucial for proving limits rigorously.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, f(x) has two definitions: 3x - 12 for x ≤ 7 and x + 2 for x > 7. Understanding how to evaluate limits at points where the function changes its definition is essential for solving the limit problem.

Epsilon-Delta Proof

An epsilon-delta proof involves demonstrating that for any chosen ε, a corresponding δ can be found that satisfies the limit definition. This method is used to show that as x approaches a certain value, the function's output can be made arbitrarily close to the limit, thus confirming the limit's existence.