How to Solve Limit Problems Involving Division by Infinity

Understanding Division by Infinity

What We're Solving:

We're exploring what happens when we divide numbers by infinity, specifically looking at 1 ÷ ∞ and ∞ ÷ ∞. This is a fascinating journey into how mathematicians handle concepts that seem to "break" normal arithmetic rules!

The Approach:

Instead of treating infinity like a regular number (which would lead us astray), we need to think about what these expressions mean as limits. Infinity isn't a number we can plug into a calculator - it's a concept that describes something growing without bound. So we'll explore what happens as numbers get infinitely large.

Step-by-Step Solution:

Step 1: Understanding 1 ÷ ∞

• Think about dividing 1 by larger and larger numbers:

- 1 ÷ 10 = 0.1 - 1 ÷ 100 = 0.01 - 1 ÷ 1,000 = 0.001 - 1 ÷ 1,000,000 = 0.000001

• Notice the pattern? As the denominator grows, the result gets closer and closer to 0
• Mathematically, we write: lim(n→∞) 1/n = 0

Step 2: Understanding ∞ ÷ ∞

• This is trickier! Let's think about it with examples:

- What if both the numerator and denominator grow at the same rate? - 100 ÷ 100 = 1 - 1,000 ÷ 1,000 = 1 - But what if they grow at different rates? - (2n) ÷ n = 2 (numerator grows twice as fast) - n ÷ (n²) = 1/n → 0 (denominator grows faster)

• This shows us that ∞ ÷ ∞ is indeterminate - it could equal any value depending on how the "infinities" compare!

Step 3: The Hierarchy Concept

• Not all infinities are equal! Some grow faster than others:

- Polynomial functions: n, n², n³... - Exponential functions: 2ⁿ, eⁿ... - The rate of growth determines the limit's behavior

The Answer:

• 1 ÷ ∞ = 0 (as a limit)
• ∞ ÷ ∞ is indeterminate - it depends on the specific functions involved and requires more advanced techniques (like L'Hôpital's Rule) to evaluate

Memory Tip:

Think of infinity like a horizon - you can always walk toward it, but you never actually reach it. When dividing by this "unreachable" concept, tiny finite numbers become negligibly small (approach 0), while comparing two "unreachables" requires looking at their relative speeds of growth!

Remember: You're not just learning arithmetic here - you're developing the mathematical thinking that underlies calculus and advanced mathematics. Great job tackling such a conceptually rich topic! 🌟