Let f(x) =x^2−2x+3.


a. For ε=0.25, find the largest value of δ>0 satisfying the statement


|f(x)−2|<ε whenever 0<|x−1|<δ.

Key Concepts

Limit Definition

The limit definition in calculus describes how a function behaves as it approaches a certain point. In this context, we are interested in the behavior of f(x) as x approaches 1. The statement |f(x)−2|<ε indicates that we want the function's value to be within ε of 2, which is the limit we are examining.

Epsilon-Delta Definition of Limit

The epsilon-delta definition formalizes the concept of limits in calculus. It states that for every ε>0, there exists a δ>0 such that if |x−c|<δ, then |f(x)−L|<ε. In this problem, c is 1 and L is 2, meaning we need to find a δ that ensures the function's output remains close to 2 when x is near 1.

Quadratic Functions

Quadratic functions are polynomial functions of the form f(x) = ax^2 + bx + c. They have a parabolic shape and can be analyzed using their vertex, axis of symmetry, and roots. In this case, f(x) = x^2 - 2x + 3 is a quadratic function, and understanding its graph helps in determining how it behaves around the point x=1.