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Visualize limit behavior by reading and interpreting function graphs
Learn step-by-step with practice exercises built right in.
Graphs provide a powerful visual way to understand limit behavior. You can often see where a function is heading!
To find from a graph:
A function has the following behavior: as x approaches 2 from the left, y approaches 4; as x approaches 2 from the right, y approaches 4; but f(2) = 1. What is ? Does it equal f(2)?
| Section | Format | Questions | Time | Weight | Calculator |
|---|---|---|---|---|---|
| Multiple Choice (No Calculator) | MCQ | 30 | 60 min | 33.3% | ๐ซ |
| Multiple Choice (Calculator) | MCQ | 15 | 45 min | 16.7% | โ |
| Free Response (Calculator) | FRQ | 2 | 30 min | 16.7% | โ |
| Free Response (No Calculator) | FRQ | 4 | 60 min | 33.3% | ๐ซ |
Avoid these 4 frequent errors
See how this math is used in the real world
A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of cm/s. How fast is the area of the circle increasing when the radius is cm?
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If the function is smooth and connected at x = a, the limit exists and equals f(a).
The curve flows smoothly through the point - both sides meet at the same spot.
Even if there's a hole at x = a, the limit can still exist!
Follow where the curve would go if the hole weren't there. Both sides still point to the same y-value.
If the left side approaches one value and the right side approaches a different value, the limit does not exist.
The curve "jumps" - there's no single value both sides agree on.
If the function shoots up to or down to , we say the limit is infinite (or DNE, depending on context).
The curve races off toward infinity - unbounded behavior.
Left-hand limit :
Right-hand limit :
Remember: If left โ right, the two-sided limit DNE!
| Feature | Symbol | Meaning |
|---|---|---|
| Solid dot | โข | Function value exists there |
| Open circle | โ | Hole - function undefined there |
| Arrow up/down | โโ | Function goes to ยฑโ |
| Break in curve | โฟ | Discontinuity |
When looking at a graph:
Imagine a graph where:
What's the limit?
(even though f(3) = 7!)
The limit is about where the function is heading, not where it is.
Finding the limit:
From the left:
From the right:
Since both one-sided limits exist and are equal:
Comparing to f(2):
We're told that f(2) = 1.
Therefore:
The limit does NOT equal the function value!
What this looks like:
From the graph of f(x), estimate lim(xโ2) f(x) if the graph shows a smooth curve passing through (2, 3).
Step 1: Understand what to look for: We need to see where the graph is heading as x approaches 2
Step 2: Trace from the left: As x moves toward 2 from the left, follow the curve to see what y-value it approaches
Step 3: Trace from the right: As x moves toward 2 from the right, follow the curve to see what y-value it approaches
Step 4: Since it's a smooth curve through (2, 3): Both left and right traces lead to y = 3
Step 5: Conclusion: lim(xโ2) f(x) = 3
Note: In this case, the limit equals f(2), which is typical for continuous functions
Answer: 3
A graph shows a hole at (3, 5) but the curve approaches 5 from both sides. What is lim(xโ3) f(x)?
Step 1: Understand the graph: โข Hole at (3, 5) means f(3) is undefined โข But the curve approaches the point from both sides
Step 2: Check left-hand limit: As x โ 3 from the left, the graph approaches y = 5
Step 3: Check right-hand limit: As x โ 3 from the right, the graph also approaches y = 5
Step 4: Apply limit definition: Since both one-sided limits equal 5: lim(xโ3) f(x) = 5
Step 5: Key insight: The limit exists even though f(3) is undefined! This is perfectly valid - limits describe nearby behavior
Answer: 5
A graph shows a jump discontinuity at x = 1. The left piece approaches 2 and the right piece approaches 4. Find lim(xโ1) f(x).
Step 1: Identify one-sided limits: โข lim(xโ1โป) f(x) = 2 (left side approaches 2) โข lim(xโ1โบ) f(x) = 4 (right side approaches 4)
Step 2: Check if they agree: Left-hand limit: 2 Right-hand limit: 4 2 โ 4
Step 3: Apply limit criterion: For lim(xโ1) f(x) to exist, left and right limits must be equal
Step 4: Conclusion: Since they're different, the two-sided limit does not exist
Step 5: Proper notation: lim(xโ1) f(x) = DNE (does not exist)
Answer: Does not exist (DNE)
A graph shows a vertical asymptote at x = 2. As x approaches 2 from the right, the graph goes up toward +โ. What is lim(xโ2โบ) f(x)?
Step 1: Identify the behavior: โข Vertical asymptote at x = 2 โข From the right (x > 2), graph increases without bound
Step 2: Interpret "increases without bound": The function values grow larger and larger They approach positive infinity
Step 3: Write the limit: lim(xโ2โบ) f(x) = โ
Step 4: Important notes: โข This is an "infinite limit" โข โ is not a real number โข We often say "the limit does not exist" โข But we can be more specific: it's unbounded positive
Step 5: Graphical meaning: Vertical asymptote where function shoots up to infinity
Answer: +โ (infinite limit)