Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx→−3 (x² − 9) / (x + 3) = −6

Verified step by step guidance

1. Understand the problem: We need to prove the limit \( \lim_{x \to -3} \frac{x^2 - 9}{x + 3} = -6 \) using the formal definition of a limit.
2. Simplify the expression: Notice that \( x^2 - 9 \) can be factored as \((x - 3)(x + 3)\). This allows us to rewrite the expression as \( \frac{(x - 3)(x + 3)}{x + 3} \). For \( x \neq -3 \), this simplifies to \( x - 3 \).
3. Apply the limit: Now, we need to find \( \lim_{x \to -3} (x - 3) \). As \( x \) approaches \(-3\), the expression \( x - 3 \) approaches \(-3 - 3 = -6\).
4. Use the formal definition of a limit: For every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x + 3| < \delta \), then \( |(x - 3) + 6| < \epsilon \).
5. Verify the condition: Simplifying \( |(x - 3) + 6| = |x + 3| \), we can choose \( \delta = \epsilon \). Therefore, for any \( \epsilon > 0 \), if \( 0 < |x + 3| < \epsilon \), then \( |x + 3| < \epsilon \), proving the limit is \(-6\).

Key Concepts

Limit Definition

The formal definition of a limit states that for a function f(x), the limit as x approaches a value 'a' is L if, for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - a| < δ, it follows that |f(x) - L| < ε. This definition is crucial for rigorously proving limit statements.

Factoring and Simplifying Expressions

To evaluate limits, especially when direct substitution results in an indeterminate form like 0/0, factoring and simplifying the expression can help. In the given limit, the expression (x² - 9) can be factored into (x - 3)(x + 3), allowing for cancellation of the (x + 3) term.

Substitution in Limits

Once the expression is simplified, substitution can be used to find the limit. In this case, after canceling the common factor, substituting x = -3 into the simplified expression yields the limit value. Understanding how to apply substitution correctly is essential for limit evaluation.