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

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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 $.

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 $.

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 $.

From $ |3(x - 7)| < \varepsilon $, we have $ |x - 7| < \frac{\varepsilon}{3} $. Thus, for $ x \leq 7 $, we can choose $ \delta = \frac{\varepsilon}{3} $.

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 $.

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Key Concepts

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

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.

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