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→1 x^4=1

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 c, for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This formalism is crucial for proving limits rigorously.

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 c. Establishing a relationship between ε and δ is essential to demonstrate that as x gets sufficiently close to c, f(x) will be within ε of L.

Polynomial Functions

Polynomial functions, such as f(x) = x^4, are continuous everywhere on their domain. This property simplifies limit calculations, as the limit of a polynomial as x approaches a point can be found by direct substitution, making it easier to apply the ε-δ definition.