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

Verified step by step guidance

1. <Step 1: Understand the problem.> We need to prove that \( \lim_{{x \to 7}} f(x) = 9 \) using the precise definition of a limit. The function \( f(x) \) is defined piecewise: \( f(x) = 3x - 12 \) for \( x \leq 7 \) and \( f(x) = x + 2 \) for \( x > 7 \).
2. <Step 2: Recall the definition of a limit.> The precise definition of a limit states that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - 7| < \delta \), then \( |f(x) - 9| < \varepsilon \).
3. <Step 3: Analyze the piecewise function.> For \( x \leq 7 \), \( f(x) = 3x - 12 \). We need \( |(3x - 12) - 9| < \varepsilon \). Simplifying, this becomes \( |3x - 21| < \varepsilon \), or \( |3(x - 7)| < \varepsilon \).
4. <Step 4: Solve for \( \delta \) for \( x \leq 7 \).> From \( |3(x - 7)| < \varepsilon \), we have \( |x - 7| < \frac{\varepsilon}{3} \). Thus, for \( x \leq 7 \), we can choose \( \delta = \frac{\varepsilon}{3} \).
5. <Step 5: Analyze the function for \( x > 7 \).> For \( x > 7 \), \( f(x) = x + 2 \). We need \( |(x + 2) - 9| < \varepsilon \). Simplifying, this becomes \( |x - 7| < \varepsilon \). Thus, for \( x > 7 \), we can choose \( \delta = \varepsilon \).

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.