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 $\underset{x\to 7}{lim}f(x)=9$ using the precise definition of a limit. The function $f(x)$ is defined piecewise: $f(x)=3x-12$ for $x\le 7$ 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 $\epsilon >0$, there exists a $\delta >0$ such that if $0<|x-7|<\delta$, then $|f(x)-9|<\epsilon$.
3. <Step 3: Analyze the piecewise function.> For $x\le 7$, $f(x)=3x-12$. We need $|(3x-12)-9|<\epsilon$. Simplifying, this becomes $|3x-21|<\epsilon$, or $|3(x-7)|<\epsilon$.
4. <Step 4: Solve for $\delta$ for $x\le 7$.> From $|3(x-7)|<\epsilon$, we have $|x-7|<\frac{\epsilon }{3}$. Thus, for $x\le 7$, we can choose $\delta =\frac{\epsilon }{3}$.
5. <Step 5: Analyze the function for $x>7$.> For $x>7$, $f(x)=x+2$. We need $|(x+2)-9|<\epsilon$. Simplifying, this becomes $|x-7|<\epsilon$. Thus, for $x>7$, we can choose $\delta =\epsilon$.

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.