Estimating Limits from Graphs

Visualize limit behavior by reading and interpreting function graphs

Reading Limits from Graphs

Graphs provide a powerful visual way to understand limit behavior. You can often see where a function is heading!

The Visual Approach

To find $\lim_{x \to a} f(x)$ from a graph:

Locate x = a on the horizontal axis
Trace from the left - Where is the curve heading as you approach a from the left?
Trace from the right - Where is the curve heading as you approach a from the right?
Check if they agree - Do both sides point to the same y-value?

Key Visual Clues

Case 1: Continuous Point

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.

Case 2: Hole (Removable Discontinuity)

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.

Case 3: Jump Discontinuity

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.

Case 4: Vertical Asymptote

If the function shoots up to $\infty$ or down to $-\infty$ , we say the limit is infinite (or DNE, depending on context).

The curve races off toward infinity - unbounded behavior.

Reading One-Sided Limits

Left-hand limit $\lim_{x \to a^-} f(x)$ :

Trace along the curve from the left
What y-value does it approach?

Right-hand limit $\lim_{x \to a^+} f(x)$ :

Trace along the curve from the right
What y-value does it approach?

Remember: If left ≠ right, the two-sided limit DNE!

Important Graph Features

| Feature | Symbol | Meaning | |---------|--------|---------| | Solid dot | • | Function value exists there | | Open circle | ○ | Hole - function undefined there | | Arrow up/down | ↑↓ | Function goes to ±∞ | | Break in curve | ⌿ | Discontinuity |

Practice Strategy

When looking at a graph:

Cover the point in question with your finger
Slide toward it from the left - where are you heading?
Slide toward it from the right - where are you heading?
If both sides agree, that's your limit!

Example Interpretation

Imagine a graph where:

As you approach x = 3 from the left, the y-values approach 5
As you approach x = 3 from the right, the y-values approach 5
But there's an open circle at (3, 5) and a solid dot at (3, 7)

What's the limit?

$\lim_{x \to 3} f(x) = 5$ (even though f(3) = 7!)

The limit is about where the function is heading, not where it is.

📚 Practice Problems

1Problem 1medium

❓ Question:

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 $\lim_{x \to 2} f(x)$ ? Does it equal f(2)?

💡 Show Solution

Finding the limit:

From the left: $\lim_{x \to 2^-} f(x) = 4$

From the right: $\lim_{x \to 2^+} f(x) = 4$

Since both one-sided limits exist and are equal:

$\lim_{x \to 2} f(x) = 4$

Comparing to f(2):

We're told that f(2) = 1.

Therefore: $\lim_{x \to 2} f(x) = 4 \neq f(2) = 1$

The limit does NOT equal the function value!

What this looks like:

There's a hole at (2, 4) where the function "wants" to be
There's a solid dot at (2, 1) where the function actually is
This is called a removable discontinuity

2Problem 2easy

❓ Question:

From the graph of f(x), estimate lim(x→2) f(x) if the graph shows a smooth curve passing through (2, 3).

💡 Show Solution

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(x→2) f(x) = 3

Note: In this case, the limit equals f(2), which is typical for continuous functions

Answer: 3

3Problem 3easy

❓ Question:

A graph shows a hole at (3, 5) but the curve approaches 5 from both sides. What is lim(x→3) f(x)?

💡 Show Solution

Step 1: Understand the graph: • Hole at (3, 5) means f(3) is undefined • But the curve approaches the point from both sides

Step 2: Check left-hand limit: As x → 3 from the left, the graph approaches y = 5

Step 3: Check right-hand limit: As x → 3 from the right, the graph also approaches y = 5

Step 4: Apply limit definition: Since both one-sided limits equal 5: lim(x→3) 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

4Problem 4medium

❓ Question:

A graph shows a jump discontinuity at x = 1. The left piece approaches 2 and the right piece approaches 4. Find lim(x→1) f(x).

💡 Show Solution

Step 1: Identify one-sided limits: • lim(x→1⁻) f(x) = 2 (left side approaches 2) • lim(x→1⁺) f(x) = 4 (right side approaches 4)

Step 2: Check if they agree: Left-hand limit: 2 Right-hand limit: 4 2 ≠ 4

Step 3: Apply limit criterion: For lim(x→1) 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(x→1) f(x) = DNE (does not exist)

Answer: Does not exist (DNE)

5Problem 5hard

❓ Question:

A graph shows a vertical asymptote at x = 2. As x approaches 2 from the right, the graph goes up toward +∞. What is lim(x→2⁺) f(x)?

💡 Show Solution

Step 1: Identify the behavior: • Vertical asymptote at x = 2 • 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(x→2⁺) f(x) = ∞

Step 4: Important notes: • This is an "infinite limit" • ∞ is not a real number • We often say "the limit does not exist" • But we can be more specific: it's unbounded positive

Step 5: Graphical meaning: Vertical asymptote where function shoots up to infinity

Answer: +∞ (infinite limit)

🎴

Practice with Flashcards

Review key concepts with our flashcard system

📖

Browse All Topics

Explore other calculus topics