Limits and One-Sided Limits in Calculus: Definitions, Properties, and Examples

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Limits in Calculus

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of functions as their inputs approach specific values. Understanding limits is essential for analyzing continuity, derivatives, and integrals.

Limit of a function: The value that a function approaches as the input approaches a particular point.
Undefined points: Some functions may be undefined at certain points, but their limits may still exist as the input nears those points.

Example Functions and Their Limits

Consider the following functions:

Domain: Both functions are undefined at .

For : is undefined because the denominator is zero.
For : is also undefined for the same reason.

To analyze their behavior near :

Key Point: The behavior of a function near a point can be different from its value at that point.

One-Sided Limits

Definition and Notation

One-sided limits describe the behavior of a function as the input approaches a point from one direction only.

Right-hand limit: is the value approaches as approaches from the right ().
Left-hand limit: is the value approaches as approaches from the left ().

If both one-sided limits exist and are equal, then the two-sided limit exists:

if and only if

Graphical Representation

Graphs can illustrate the difference between left-hand and right-hand limits, especially at points of discontinuity.

If approaches different values from the left and right, the two-sided limit does not exist at that point.

Absolute Value Functions and Limits

Definition of Absolute Value

The absolute value of a real number is defined as:

For expressions involving absolute value, such as :

Evaluating Limits with Absolute Value

When evaluating limits involving absolute value, consider the direction from which approaches the point.

For , since , , so .
For , since , , so .

Example:

Since the left and right limits are not equal, the two-sided limit does not exist:

does not exist (DNE).

Infinite Limits

Types of Infinite Limits

Infinite limits describe the behavior of functions as they increase or decrease without bound near a particular point.

Infinite limits from the left: If increases without bound as , then . If decreases without bound, .
Infinite limits from the right: If increases without bound as , then . If decreases without bound, .
Two-sided infinite limits: If increases without bound as , then . If decreases without bound, .

Type	Condition	Limit Notation	Result
Infinite limit from left	, increases without bound
Infinite limit from left	, decreases without bound
Infinite limit from right	, increases without bound
Infinite limit from right	, decreases without bound
Two-sided infinite limit	, increases without bound
Two-sided infinite limit	, decreases without bound

Graphical Examples of Infinite Limits

Graphs can show functions approaching infinity or negative infinity as approaches a specific value from the left or right.

For ,
For ,
For ,
For ,

Summary Table: Types of Limits

Type of Limit	Notation	Description
Two-sided limit		Approaches from both sides
Right-hand limit		Approaches from the right ()
Left-hand limit		Approaches from the left ()
Infinite limit		Function increases or decreases without bound

Key Takeaways

Limits describe the behavior of functions near specific points, even if the function is undefined at those points.
One-sided limits are crucial for analyzing discontinuities and piecewise functions.
Absolute value functions require careful consideration of the direction of approach.
Infinite limits indicate unbounded behavior near certain points.