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→3 x^3=27

Verified step by step guidance

1. Step 1: Recall the precise definition of a limit. For the limit \( \lim_{x \to 3} x^3 = 27 \) to hold, for every \( \varepsilon > 0 \), there must exist a \( \delta > 0 \) such that if \( 0 < |x - 3| < \delta \), then \( |x^3 - 27| < \varepsilon \).
2. Step 2: Express \( |x^3 - 27| \) in terms of \( x - 3 \). Notice that \( x^3 - 27 = (x - 3)(x^2 + 3x + 9) \). Therefore, \( |x^3 - 27| = |x - 3| |x^2 + 3x + 9| \).
3. Step 3: To find a suitable \( \delta \), we need to bound \( |x^2 + 3x + 9| \) when \( x \) is close to 3. Assume \( |x - 3| < 1 \), which implies \( 2 < x < 4 \).
4. Step 4: Estimate \( |x^2 + 3x + 9| \) for \( 2 < x < 4 \). Calculate the maximum value of \( x^2 + 3x + 9 \) in this interval. For example, at \( x = 2 \), \( x^2 + 3x + 9 = 19 \), and at \( x = 4 \), \( x^2 + 3x + 9 = 37 \). Thus, \( |x^2 + 3x + 9| \leq 37 \).
5. Step 5: Choose \( \delta = \min(1, \varepsilon/37) \). This ensures that \( |x - 3| < \delta \) implies \( |x^3 - 27| = |x - 3| |x^2 + 3x + 9| < \varepsilon \), satisfying the definition of the limit.

Key Concepts

Limit Definition

The precise definition of a limit states that for a function f(x) to approach a limit L as x approaches a value a, for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. This formalism is essential for rigorously proving limits in calculus.

Epsilon-Delta Relationship

In the context of limits, the ε (epsilon) represents how close f(x) must be to the limit L, while δ (delta) represents how close x must be to the point a. Establishing a relationship between ε and δ is crucial for demonstrating that as x gets sufficiently close to a, f(x) will be within ε of L, thus proving the limit exists.

Polynomial Functions

Polynomial functions, such as f(x) = x^3, are continuous everywhere on their domain. This continuity implies that the limit of a polynomial as x approaches any point can be found by direct substitution. Understanding this property simplifies the process of proving limits for polynomial functions.