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Continuity and Limits at Infinity

Understanding unbroken curves and behavior at the edges of the universe.

Once we understand basic limits, we encounter two massive questions: "When is a line unbroken?" and "What happens if we keep going forever?" These concepts are known as Continuity and Limits at Infinity.

1. What is Continuity?

Intuitively, a function is continuous if you can draw its graph without lifting your pen from the paper. There are no holes, no jumps, and no breaks.

[Image of continuous vs discontinuous graph]

Mathematically, for a function f(x) to be continuous at a specific point c, three conditions must be met:

1. f(c) is defined: The point exists.
2. Limit exists: The left and right limits match.
3. Limit equals value: The limit as x approaches c is actually equal to f(c).

Types of Discontinuities

When a function is not continuous, it has a "discontinuity". Common types include:

  • Removable (Hole): The limit exists, but the point is missing or wrong. (Like a bridge with one missing plank).
  • Jump Discontinuity: The graph jumps from one height to another instantly. (Like a step function).
  • Infinite Discontinuity: The graph shoots up to positive or negative infinity (Vertical Asymptote).

2. Limits at Infinity

What happens to a function when x gets really, really big? This is called looking at the "End Behavior" of the graph.

Consider the function f(x) = 1/x. If x is 10, y is 0.1. If x is 1,000, y is 0.001. If x is 1,000,000, y is 0.000001.

[Image of graph of 1/x showing horizontal asymptote]

As x approaches infinity, y gets closer and closer to zero, but never quite touches it. We write this as:

lim (x→∞) (1/x) = 0

This creates a Horizontal Asymptote at y = 0.

3. Infinite Limits vs. Limits at Infinity

These two terms sound similar but mean different things:

  • Limits AT Infinity: x is going to infinity (looking far right on the graph). The result is usually a specific number (Horizontal Asymptote).
  • Infinite Limits: x is approaching a specific number, but the result explodes to infinity (Vertical Asymptote).

4. The Rule of Dominance

When calculating limits at infinity for complex polynomials, a useful trick is to look at the "dominant term" (the one with the highest power).

Example: lim (x→∞) (3x² + 5x) / (2x² - 1)

For huge x, the x² terms dominate. The small x and numbers don't matter.
≈ 3x² / 2x²
= 3/2 = 1.5

This tells us the graph eventually flattens out at the height of 1.5.