Finding Limits and Function Values from a Graph- Worked Examples

What This Article Covers

You're looking at a graph and your professor asks: "What's the limit as x approaches 3?" and "What's f(3)?" These are two different questions. Your graph can answer both—if you know how to read it.

This guide walks you through finding limits and function values directly from graphs. No complicated formulas. Just visual reasoning with worked examples you can replicate on your next exam.

Function Value vs. Limit: Know the Difference First

These two concepts trip up students constantly. They're related, but they're not the same thing.

Function Value: f(a)

The function value at point a is simply the y-coordinate where the graph crosses x = a. It must exist on the actual curve. If there's a hole, jump, or the point simply doesn't exist—you don't have a function value there.

Limit: lim(x→a) f(x)

The limit describes what y-value the function is approaching as x gets arbitrarily close to a. This can exist even when f(a) doesn't exist. The limit only cares about the behavior near the point, not at the point itself.

Think of it this way: the limit is about the neighborhood. The function value is about the specific address.

How to Find Limits from a Graph

Finding limits visually comes down to asking one question: What is the y-value getting close to as I trace the graph toward my x-value from both sides?

Step-by-Step Process

• Locate your target x-value on the horizontal axis
• Trace the graph from the left toward that x-value—note the y-value you're approaching
• Trace the graph from the right toward that x-value—note the y-value you're approaching
• If both sides approach the same y-value, that is your limit
• If the sides approach different y-values, the limit does not exist (DNE)

Key Graphical Features That Affect Limits

1. Holes (Point Discontinuities)

A hole means the function isn't defined at that single point, but the limit can still exist. The graph approaches a y-value but doesn't reach it there.

2. Jumps (Jump Discontinuities)

The graph suddenly jumps from one y-value to another. If the left and right sides don't meet, the limit DNE—even if both sides have well-defined y-values.

3. Vertical Asymptotes

The graph shoots off toward infinity. Limits at vertical asymptotes typically DNE, though sometimes one-sided limits exist.

4. Cusps and Sharp Turns

The function exists but has a corner. The limit may still exist if both sides approach the same value.

Worked Examples

Example 1: Limit Exists, Function Value Exists

Given: The graph shows a continuous curve passing through (2, 4).

Find: lim(x→2) f(x) and f(2)

Solution:

Since the curve is continuous at x = 2 and passes through (2, 4):

• f(2) = 4 (the y-value at x = 2)
• lim(x→2) f(x) = 4 (the graph approaches 4 from both sides)

In continuous cases, the limit and function value are identical. This is the easy scenario.

Example 2: Hole in the Graph

Given: The graph shows a curve approaching y = 3 from both sides at x = 1, but there's an open circle (hole) at (1, 5).

Find: lim(x→1) f(x) and f(1)

Solution:

• lim(x→1) f(x) = 3 (the y-value both sides approach)
• f(1) = Does Not Exist (there's a hole at x = 1)

The graph gets arbitrarily close to y = 3, but never reaches it at x = 1. The limit is 3. The function value is undefined.

Example 3: Different Left and Right Limits

Given: At x = 3, the graph approaches y = 2 from the left and y = 6 from the right. There's a closed circle at (3, 4).

Find: lim(x→3) f(x) and f(3)

Solution:

• lim(x→3) f(x) = Does Not Exist (left-hand limit ≠ right-hand limit)
• f(3) = 4 (the closed circle at (3, 4) tells us the function value)

The limit DNE because the graph doesn't settle on a single y-value. But the function value is clearly defined at that point—these are completely independent questions.

Example 4: Vertical Asymptote

Given: The graph shows a vertical asymptote at x = -1. As x approaches -1 from the left, y goes to +∞. From the right, y goes to -∞.

Find: lim(x→-1) f(x)

Solution:

• lim(x→-1) f(x) = Does Not Exist

The function blows up in opposite directions on each side. No single y-value captures the behavior, so the limit DNE.

Example 5: One-Sided Limits

Given: At x = 4, the graph approaches y = 7 from the left and continues to y = 7 from the right, but there's a hole at (4, 7) and a closed point at (4, 3).

Find: lim(x→4) f(x), lim(x→4⁻) f(x), lim(x→4⁺) f(x), and f(4)

Solution:

• lim(x→4⁻) f(x) = 7 (approaching from left)
• lim(x→4⁺) f(x) = 7 (approaching from right)
• lim(x→4) f(x) = 7 (both one-sided limits match)
• f(4) = 3 (the closed point is the actual function value)

This is a removable discontinuity. The limit exists and equals 7, but the function value is 3. They don't match.

Quick Reference Table

Getting Started: Your Checklist for Any Graph Problem

When you're staring at a graph on a test, run through this mentally:

• Step 1: Identify the x-value you're evaluating
• Step 2: For limits: trace from left, trace from right. Do they agree?
• Step 3: For function values: is there a point directly on the graph at that x-value? Look for closed circles, not open circles
• Step 4: Write your answer. Don't assume the limit equals the function value—they often don't

Common Mistakes to Avoid

• Reading the wrong point: Confusing an open circle (doesn't exist) with a closed circle (exists). This is the #1 error.
• Assuming limit = function value: They match only when the function is continuous. Don't assume continuity.
• Ignoring one-sided behavior: Always check both sides. A limit can fail to exist even when each side individually looks fine.
• Misreading axes: Check the scale. A tiny-looking jump might represent a huge change in y.

Bottom Line

Limits and function values from graphs come down to two separate questions. Limits ask about approach behavior. Function values ask what's actually there. They can match, differ, or have one exist while the other doesn't.

Read the graph carefully. Check both sides. Look for open versus closed circles. That's it.