No, 1/infinity is not zero, but it approaches zero.
Understanding the Concept
The core issue lies in the fact that "infinity" is not a number. You can't treat it as a standard operand in mathematical operations like division. Think of infinity as a concept representing something without any bound.
Why it Approaches Zero
While you can't directly divide by infinity, we can consider the behavior of 1/x as x gets larger and larger, approaching infinity. This is where the concept of a limit comes into play.
- Limits: In calculus, we use limits to describe what value a function approaches as its input approaches some value (like infinity). We write this as: lim (x→∞) 1/x = 0. This reads: "The limit of 1/x as x approaches infinity is 0."
- The Meaning: This doesn't mean 1/∞ equals 0. It means that as x gets arbitrarily large, 1/x gets arbitrarily close to 0. It's a process, not a direct calculation.
Example:
Consider the following:
- 1/1 = 1
- 1/10 = 0.1
- 1/100 = 0.01
- 1/1000 = 0.001
- 1/1000000 = 0.000001
As the denominator increases (approaching infinity), the result gets closer and closer to zero.
In Summary
- Infinity isn't a number you can directly use in division.
- The expression 1/infinity is more accurately understood as the limit of 1/x as x approaches infinity.
- This limit is 0, meaning 1/x gets arbitrarily close to 0 as x gets arbitrarily large.
