The answer is indeterminate. While it might seem intuitive that infinity multiplied by zero would equal zero, in the realm of mathematics, it's considered an indeterminate form.
Understanding Indeterminate Forms
An indeterminate form arises in the context of limits. It means that knowing the limits of two functions individually isn't enough to determine the limit of their product, quotient, or other combination. The actual result depends on the specific functions involved.
Why Isn't It Simply Zero?
The confusion often stems from thinking of "infinity" as a concrete number. Infinity isn't a number; it's a concept representing something without any bound. When you're dealing with limits, expressions like "infinity" and "zero" represent values that functions are approaching, not fixed quantities.
Examples Illustrating Indeterminacy
Consider these examples using limits:
-
Limit 1: lim (x→∞) x * (1/x) = 1
- Here, as x approaches infinity,
xgoes to infinity and1/xapproaches zero. However, their product approaches 1.
- Here, as x approaches infinity,
-
Limit 2: lim (x→∞) x^2 * (1/x) = ∞
- In this case,
x^2goes to infinity faster than1/xapproaches zero. So, the product goes to infinity.
- In this case,
-
Limit 3: lim (x→∞) x * (1/x^2) = 0
- Here,
1/x^2approaches zero faster thanxgoes to infinity, resulting in a product that approaches zero.
- Here,
These examples demonstrate that the expression ∞ * 0 can approach different values depending on the specific functions involved. Therefore, it is considered an indeterminate form.
The Case of Exact Zero
If you are multiplying infinity by the exact quantity zero (not something approaching zero), then the answer is zero. However, this is rarely the context in mathematical analysis. The issue arises when dealing with limits, where we are considering values approaching infinity and zero, not fixed quantities.
Other Indeterminate Forms
Besides 0 * ∞, other indeterminate forms include:
- 0/0
- ∞/∞
- ∞ - ∞
- 1∞
- 00
- ∞0
