Limits

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of a function as its input approaches a particular value. Understanding limits is essential for studying continuity, derivatives, and integrals.

• Limit: The value that a function approaches as the input (x) approaches a specific point.

• Undefined Points: Some functions are not defined at certain points, but their limits may still exist at those points.

Example Functions and Their Behavior Near Undefined Points

Consider the following functions, both undefined at x = 2:

Both functions are undefined at . The study of their limits as approaches 2 reveals important information about their behavior near this point.

One-Sided Limits

Definition of One-Sided Limits

One-sided limits describe the behavior of a function as the input approaches a specific value from one side only—either from the left or the right.

• Right-Hand Limit: The value a function approaches as approaches from values greater than (from the right).

• Left-Hand Limit: The value a function approaches as approaches from values less than (from the left).

If the value of approaches a real number as approaches from the right (), then:

If the value of approaches a real number as approaches from the left (), then:

Definition of a Two-Sided Limit

A two-sided limit exists if and only if both the left-hand and right-hand limits exist and are equal:

• if

Example: Evaluating One-Sided Limits

Evaluate the following one-sided limit:

To solve, consider the definition of the absolute value and analyze the function as approaches 3 from the left and right.

Infinite Limits

Types of Infinite Limits

Infinite limits describe the behavior of a function as it increases or decreases without bound near a certain point. There are three main types:

• Infinite Limits from the Left: If increases without bound as , then . If decreases without bound as , then .

• Infinite Limits from the Right: If increases without bound as , then . If decreases without bound as , then .

• Two-Sided Infinite Limits: If increases without bound as , then . If decreases without bound as , then .

Summary Table: Types of Infinite Limits

Key Points

• Limits help describe the behavior of functions near points of discontinuity or undefined values.

• One-sided limits are essential for understanding piecewise and absolute value functions.

• Infinite limits indicate vertical asymptotes or unbounded behavior near certain points.

Example Application

For , as approaches 2 from the right, increases without bound, so . As approaches 2 from the left, decreases without bound, so .