What is a Limit?

An intuitive introduction to the concept of limits in calculus

Understanding Limits Intuitively

A limit describes what happens to a function as the input gets closer and closer to a certain value. Think of it as asking: "Where is this function heading?"

Key Idea: Limits are about the journey, not the destination. We care about where a function is going, not necessarily where it is.

The Limit Notation

We write:

$\lim_{x \to a} f(x) = L$

This reads: "The limit of f(x) as x approaches a equals L"

What This Means

$x \to a$ : The variable x gets closer and closer to the value a
$f(x) \to L$ : The function values get closer and closer to L
We don't care about what happens at x = a, only what happens near it

A Simple Example

Consider the function: $f(x) = 2x + 1$

What is $\lim_{x \to 3} (2x + 1)$ ?

As x gets close to 3:

When x = 2.9: $f(2.9) = 2(2.9) + 1 = 6.8$
When x = 2.99: $f(2.99) = 2(2.99) + 1 = 6.98$
When x = 2.999: $f(2.999) = 2(2.999) + 1 = 6.998$

The values are getting closer to 7!

Therefore: $\lim_{x \to 3} (2x + 1) = 7$

Why Limits Matter

Limits are the foundation of calculus because they help us:

Define derivatives (rate of change)
Define integrals (area under curves)
Understand function behavior near problematic points
Handle infinity and infinitesimal quantities

Important Note

The limit of f(x) as x approaches a might be different from f(a), or f(a) might not even exist!

📚 Practice Problems

1Problem 1easy

❓ Question:

Using your intuition, estimate $\lim_{x \to 2} (3x - 1)$ by checking values near x = 2.

💡 Show Solution

Let's check values approaching 2 from both sides:

From the left (x < 2):

x = 1.9: $f(1.9) = 3(1.9) - 1 = 4.7$
x = 1.99: $f(1.99) = 3(1.99) - 1 = 4.97$
x = 1.999: $f(1.999) = 3(1.999) - 1 = 4.997$

From the right (x > 2):

x = 2.1: $f(2.1) = 3(2.1) - 1 = 5.3$
x = 2.01: $f(2.01) = 3(2.01) - 1 = 5.03$
x = 2.001: $f(2.001) = 3(2.001) - 1 = 5.003$

Both sides approach 5, so: $\lim_{x \to 2} (3x - 1) = 5$

2Problem 2easy

❓ Question:

What does lim(x→2) f(x) = 5 mean in words?

💡 Show Solution

Step 1: Break down the notation: • lim means "limit" • x→2 means "as x approaches 2" • f(x) is the function • = 5 is the value the function approaches

Step 2: Interpret: As x gets closer and closer to 2 (from both sides), the function values f(x) get closer and closer to 5

Step 3: Important note: This does NOT require f(2) = 5 The limit describes behavior NEAR x = 2, not AT x = 2

Answer: As x approaches 2, f(x) approaches 5

3Problem 3easy

❓ Question:

If f(x) = (x² - 4)/(x - 2), describe what happens to f(x) as x approaches 2.

💡 Show Solution

Step 1: Note that f(2) is undefined: f(2) = (4 - 4)/(2 - 2) = 0/0 (indeterminate)

Step 2: Factor the numerator: f(x) = (x² - 4)/(x - 2) = [(x - 2)(x + 2)]/(x - 2)

Step 3: Simplify for x ≠ 2: f(x) = x + 2

Step 4: Evaluate the limit: As x→2, f(x) = x + 2 → 2 + 2 = 4

Step 5: Create a table to verify: x: 1.9 1.99 1.999 →2← 2.001 2.01 2.1 f(x): 3.9 3.99 3.999 4 4.001 4.01 4.1

Answer: As x approaches 2, f(x) approaches 4

4Problem 4medium

❓ Question:

Consider f(x) = |x|/x. What is lim(x→0) f(x)?

💡 Show Solution

Step 1: Analyze from the left (x < 0): When x < 0, |x| = -x f(x) = -x/x = -1

Step 2: Analyze from the right (x > 0): When x > 0, |x| = x f(x) = x/x = 1

Step 3: Check if left and right limits agree: lim(x→0⁻) f(x) = -1 lim(x→0⁺) f(x) = 1

Step 4: Since left ≠ right: The limit does NOT exist

Step 5: Conclusion: lim(x→0) f(x) does not exist (DNE)

Answer: The limit does not exist

5Problem 5hard

❓ Question:

Explain why a function can have a limit at x = a even if f(a) is undefined.

💡 Show Solution

Step 1: Understand what a limit measures: A limit describes the behavior of f(x) as x APPROACHES a, not the value AT x = a

Step 2: Example: f(x) = (x² - 1)/(x - 1) • f(1) is undefined (0/0) • But lim(x→1) f(x) = 2

Step 3: Why this works: Factor: f(x) = [(x - 1)(x + 1)]/(x - 1) = x + 1 for x ≠ 1 As x→1, f(x)→2, even though f(1) doesn't exist

Step 4: The key concept: Limits care about NEARBY values, not the value AT the point We can get arbitrarily close to x = a without ever reaching it

Step 5: Practical meaning: This is why limits are useful for defining derivatives (which often involve 0/0 forms)

Answer: Limits describe nearby behavior, not the value at the point itself

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