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Limits

Overview

Approaching the edge.

What is it?

The value a function approaches as the input gets closer and closer to some number.

History

Formalized by Weierstrass to solve Zeno's Paradox: how can you move anywhere if you always have to go halfway first?

Key Idea

Just plug the number in. If you get 0/0, do some Algebra (factor), then plug in again.

Practice This Topic

Concept Guide

Plain English: A limit asks 'Where are we going?', not 'Where are we?'. It allows us to handle 'forbidden' math, like dividing by zero, by seeing what happens when we get infinitesimally close to the danger zone.

Real-world example: Speedometers. A car's speed at an exact instant is a limit. It's distance divided by time, where the time elapsed shrinks to zero.

How to do it

  1. Direct Substitution: Plug the target x-value into the function.
  2. If you get a real number, you are done.
  3. If you get 0/0 (Indeterminate Form), you must simplify.
  4. Factor the top and bottom, cancel common terms, then plug the number in again.
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Common Pitfall

Confusing the Limit with the Value. A function can have a hole at x=2 (undefined), but still have a perfectly valid limit as you approach 2.

Word Problem
"A factory's production efficiency is modeled by f(x) = (x^2 - 9) / (x - 3), where x is the number of hours the machine runs. What is the efficiency approaching as x gets closer to 3 hours?"
Reasoning: Plugging in 3 gives 0/0. Factor the top: (x-3)(x+3). Cancel the (x-3)s. You are left with x+3. Plug in 3 again. The limit is 6.

Practice Examples

Notation
2 lim x = 4 x->2
As x gets close to 2, x^2 gets close to 4.