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→a (mx+b)=ma+b, for any constants a, b, and m

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 crucial for proving limits rigorously.

Linear Functions

The expression mx + b represents a linear function, where m is the slope and b is the y-intercept. Understanding the behavior of linear functions as x approaches a specific value is essential for evaluating limits, as they exhibit predictable and continuous behavior.

Epsilon-Delta Relationship

In limit proofs, establishing a relationship between ε and δ is vital. For the limit lim x→a (mx + b) = ma + b, one can show that choosing δ = ε/|m| ensures that the condition |f(x) - L| < ε is satisfied, thereby proving the limit exists.