Limits and Graphs

Part 1: Understanding Limits from a Graph

Limits describe the behavior of a function as the input approaches a particular value. They are fundamental in calculus for analyzing continuity and change.

• Definition of a Limit: The limit of a function as approaches is written as: This means the values of the graph are approaching as gets close to from both sides.

One-Sided Limits

• Left-hand limit: Approach from the left.

• Right-hand limit: Approach from the right.

Two-Sided Limits

• Definition: exists only if both one-sided limits exist and are equal:

Function Value vs. Limit

• Function value: is the actual value at .

• Limit: is what the function approaches as nears .

• Note: These can be different!

How to Read a Graph for Limits

• Follow arrows or lines to see how the graph approaches a point.

Example Graph-Based Problems

• Problem 1: From left and right, graph approaches 3. Answer:

• Problem 2: Left approaches 4, right approaches 5. Answer: Limits do not match, so limit does not exist.

• Problem 3: Look only to the left of , graph approaches 5. Answer:

• Problem 4: Left and right both approach 4. Answer:

• Problem 5: Both sides approach 3. Answer:

Part 2: Limits from Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. Limits at boundaries require careful analysis of each piece.

Step-by-Step for Limits

1. Use the correct piece for the interval as approaches the boundary.

2. Compute left and right limits separately.

3. If both sides agree, the limit exists; if not, it does not exist.

Function Value

• Function value at a boundary may differ from the limit.

Matching Graphs

• Open circle: not defined at

• Closed dot: defined at

• Graph jumps: limit does not exist

Summary: How to Solve These Problems

1. Identify if you’re finding a left-hand, right-hand, or two-sided limit.

2. For piecewise, plug into correct piece based on direction.

3. For graphs, trace the curve to see what value is approached.

4. Don’t confuse the limit with the function’s actual value.

5. If left ≠ right, the limit does not exist.

What is a Piecewise Function?

A piecewise function is defined by different expressions over different intervals of the domain.

• Example:

Steps to Graph a Piecewise Function

1. Identify intervals and functions.

2. Use test points in each interval.

3. Open or closed dots:

   • Solid dot: interval includes the point

   • Open circle: interval excludes the point

4. Plot the pieces and connect smoothly where possible.

How to Evaluate Limits from a Piecewise Function

1. Identify which sides you’re approaching from.

2. Plug into the correct piece for left/right-hand limits.

3. Compare one-sided limits. If they agree, the limit exists; if not, it does not exist.

Recognizing Discontinuities

A function is discontinuous at a point if:

• The limit doesn’t exist.

• The function value doesn’t match the limit.

Types:

• Jump Discontinuity: Left and right limits exist but aren’t equal.

• Removable Discontinuity: Left and right limits agree, but function is not defined at that point.

Example Walkthroughs

• Two-piece limit at a point: Left: Right: Limit does not exist (DNE)

• Three-part piecewise function (with jump): Left: Right: Limit does not exist (DNE)

Checklist When Solving These Problems

• Identify the correct piece of the function based on the limit’s direction.

• Evaluate left and right-hand limits separately.

• Both limits must agree for limit to exist.

• Open circle = excluded, solid dot = included.

• Use values close to the boundary to understand function behavior.

• Don’t confuse function value with limit.

Practice Tips

• Sketch graphs by hand to understand behavior.

• Practice determining limit values even if the function has jumps or holes.

• Use test values just before and after the point.

Summary Table

Additional info: This guide covers topics from "Limits and Continuity" and "Graphical Applications of Derivatives" in the Calculus curriculum, focusing on graphical and piecewise approaches to limits and discontinuities.