The Journey to the Edge: Understanding Limits

Imagine you’re walking toward a door. You take one step. Then half a step. Then a quarter step. Then an eighth… You keep getting closer and closer, but you never quite touch the door. That’s what a limit is! It’s about understanding what happens as you get infinitely close to something.

What is a Limit? (The Big Picture)

Think of a limit like a treasure hunt. You follow clues that get you closer and closer to the treasure chest. The limit is what’s inside the chest—even if you never actually open it!

In math, we write:

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

This means: “As x gets super close to a, f(x) gets super close to L.”

Limits Intuition

The Walking Ant

Picture a tiny ant walking along a number line toward the number 3.

• First, the ant is at 2.5
• Then at 2.9
• Then at 2.99
• Then at 2.999…

The ant keeps getting closer to 3 but never quite reaches it. We ask: “What number is the ant approaching?”

That’s a limit!

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

As x approaches 2:

• When x = 1.9 → 3(1.9) + 1 = 6.7
• When x = 1.99 → 3(1.99) + 1 = 6.97
• When x = 1.999 → 3(1.999) + 1 = 6.997

The answer is getting closer and closer to 7!

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

Limit Definition

The Formal Promise

The precise definition uses two Greek letters: epsilon (ε) and delta (δ).

Think of it like a game:

1. Your friend says “Get within ε meters of the target”
2. You figure out you need to stay within δ meters of the starting point
3. If you can always win this game (for any ε they choose), the limit exists!

Formal Definition: $\lim_{x \to a} f(x) = L$ means:

For every tiny ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε

Translation: Get as close as you want to L by getting close enough to a!

graph TD
    A["Pick any small ε"] --> B["Find the right δ"]
    B --> C["Stay within δ of a"]
    C --> D["f#40;x#41; stays within ε of L"]
    D --> E["Limit exists!"]

One-Sided Limits

The Left Door and the Right Door

Sometimes it matters which direction you approach from—like entering a room from the left hallway or the right hallway.

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

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

The Secret Rule

A two-sided limit exists only if both one-sided limits:

1. Exist
2. Are equal to each other

Example: Consider f(x) = |x|/x

• From the left (x → 0⁻): f(x) = -1
• From the right (x → 0⁺): f(x) = +1

Since -1 ≠ +1, the limit at 0 does not exist!

graph TD
    A["Approaching x = a"] --> B{From which side?}
    B -->|Left side| C["x → a⁻"]
    B -->|Right side| D["x → a⁺"]
    C --> E{Both equal?}
    D --> E
    E -->|Yes| F["Limit exists"]
    E -->|No| G["Limit DNE"]

Limit Laws

The Math Toolkit

Limits follow friendly rules—like LEGO blocks that snap together!

Example: Find $\lim_{x \to 3} (2x^2 + 5x - 1)$

Using the laws:

• = 2·(lim x²) + 5·(lim x) - 1
• = 2·(3)² + 5·(3) - 1
• = 18 + 15 - 1
• = 32

Squeeze Theorem

The Sandwich Rule

Imagine you’re a cookie squeezed between two pieces of bread. If both bread slices move toward your mouth, the cookie must follow!

The Theorem: If g(x) ≤ f(x) ≤ h(x) near a point, and: $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$

Then: $\lim_{x \to a} f(x) = L$ too!

Famous Example: Prove $\lim_{x \to 0} x \cdot \sin(1/x) = 0$

We know: -1 ≤ sin(1/x) ≤ 1

Multiply by |x|: -|x| ≤ x·sin(1/x) ≤ |x|

As x → 0:

• lim(-|x|) = 0
• lim(|x|) = 0

Squeezed! So lim x·sin(1/x) = 0

graph TD
    A["Bottom function g#40;x#41;"] --> D["Both approach L"]
    B["Mystery function f#40;x#41;"] --> E["Squeezed to L!"]
    C["Top function h#40;x#41;"] --> D
    D --> E

Limits at Infinity

What Happens at the End of the Universe?

When x grows without bound, where does f(x) go?

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

means: as x gets bigger and bigger, f(x) approaches L.

Key Patterns

Pattern 1: $\lim_{x \to \infty} \frac{1}{x^n} = 0$ (any n > 0)

As x gets huge, 1 divided by a huge number shrinks to 0!

Pattern 2: For rational functions, divide everything by the highest power:

Example: $\lim_{x \to \infty} \frac{3x^2 + 2x}{5x^2 - 1}$

Divide top and bottom by x²: $= \lim_{x \to \infty} \frac{3 + 2/x}{5 - 1/x^2} = \frac{3 + 0}{5 - 0} = \frac{3}{5}$

The Race of Powers

Infinite Limits

When Numbers Explode!

Sometimes f(x) doesn’t approach a number—it blasts off to infinity!

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

means: as x approaches a, f(x) grows without bound.

Example: $\lim_{x \to 0} \frac{1}{x^2} = \infty$

Let’s see:

• x = 0.1 → 1/0.01 = 100
• x = 0.01 → 1/0.0001 = 10,000
• x = 0.001 → 1/0.000001 = 1,000,000

The values are exploding upward!

Watch the Signs

Asymptotes

The Invisible Walls

Asymptotes are lines that a graph gets infinitely close to but never actually touches—like a moth circling a flame!

Three Types

1. Vertical Asymptote (x = a) Where the function explodes to infinity.

• Find where the denominator = 0
• Check that numerator ≠ 0 there

Example: f(x) = 1/(x - 2) has a vertical asymptote at x = 2

2. Horizontal Asymptote (y = L) Where the function settles as x → ±∞.

• Calculate $\lim_{x \to \infty} f(x)$

Example: f(x) = 3x/(x + 1)

• $\lim_{x \to \infty} = 3/1 = 3$
• Horizontal asymptote: y = 3

3. Oblique (Slant) Asymptote When numerator degree is exactly 1 more than denominator.

• Divide polynomials to find the line!

Example: f(x) = (x² + 2x)/(x + 1)

• Divide: x² + 2x ÷ (x + 1) = x + 1 - 1/(x+1)
• Oblique asymptote: y = x + 1

graph TD
    A["Find Asymptotes"] --> B{Type?}
    B --> C["Vertical: Set denominator = 0"]
    B --> D["Horizontal: Limit as x → ∞"]
    B --> E["Oblique: Polynomial division"]

Putting It All Together

Limits are the foundation of calculus. They help us:

• Define derivatives (instantaneous rates)
• Define integrals (areas under curves)
• Understand function behavior at edges

Quick Summary

You Did It!

You now understand the fundamental concept that unlocks all of calculus. Limits are like a superpower—they let you peek at values you can’t directly calculate. Every derivative, every integral, every slope of a tangent line… it all starts with limits.

Go forth and limit!