In Exercises 1-6, sketch the interval (a, b) on the x -axis...

In Exercises $1-6,$ sketch the interval $(a, b)$ on the $x$ -axis with the point $x_{0}$ inside. Then find a value of $\delta > 0$ such that for all $ x, 0 < \left|x-x_{0}\right| < \delta \Rightarrow a < x < b$
$$
a=-7 / 2, \quad b=-1 / 2, \quad x_{0}=-3 / 2
$$

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Key Concepts

Distance to Endpoints

The distance from a point to the endpoints of an interval is the smallest distance between that point and each endpoint. In problems involving ?-neighborhoods within an interval, ? is often chosen to be the minimum of these distances to ensure that the neighborhood is entirely contained within the interval.

Absolute Value Inequality

An absolute value inequality describes the distance between two real numbers without considering direction. In the context of ?-neighborhoods, the inequality 0 < |x - x?| < ? defines all points x that are within ? units of x?, while excluding the point x? itself. This form is frequently used in the epsilon-delta definitions of limits.

Open Interval

An open interval is a set of real numbers that includes all numbers between two endpoints but not the endpoints themselves. Understanding open intervals is crucial in real analysis when discussing continuity and limits, as it defines a set in which every point is surrounded by other points that also belong to the set.

?-Neighborhood

A ?-neighborhood of a point, typically in the context of epsilon-delta definitions in calculus, is a set of points whose distance from the point is less than ?. This concept is essential for rigorously describing 'closeness' around a point, which underpins the formal definitions of limits and continuity.

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