No, a number divided by infinity is not always 0.
While intuitively it might seem that way, the concept of "infinity" requires a more nuanced understanding, especially when dealing with mathematical operations like division. The answer depends on the context and how "division by infinity" is interpreted.
Here's a breakdown:
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Limits and Approaching Infinity: In calculus and analysis, we often deal with limits. If we consider a constant number c divided by a variable x that approaches infinity, then the limit of c/x as x approaches infinity is indeed 0. This is often written as:
lim (x→∞) c/x = 0
In this sense, dividing a constant by a quantity approaching infinity approaches zero.
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Indeterminate Forms: However, the expression "division by infinity" can arise in more complex situations, such as evaluating limits of the form ∞/∞ or 0/0. These are called indeterminate forms. L'Hôpital's rule is frequently used to evaluate these limits, and the result is not necessarily 0. The limit could be any real number, infinity, or may not exist.
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Example 1 (∞/∞): Consider the limit of x2 / x as x approaches infinity. Both the numerator and denominator approach infinity. Applying L'Hôpital's Rule, we differentiate the numerator and denominator to get 2x/1. The limit of 2x as x approaches infinity is infinity. Therefore, the original limit is also infinity, not 0.
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Example 2 (0/0): Consider the limit of x/x2 as x approaches 0. Both the numerator and denominator approach zero. Applying L'Hôpital's Rule, we get 1/(2x). The limit as x approaches 0 is infinity, not 0.
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Division by Zero: A related, but distinct, issue is division by zero. Infinity is not a real number. Division by zero is undefined in standard arithmetic and results in undefined results, not infinity.
In summary:
| Concept | Result | Explanation |
|---|---|---|
| limit of c/x as x approaches infinity | 0 | Where c is a constant. |
| Indeterminate forms (∞/∞, 0/0) | Not always 0 | Requires evaluation using techniques like L'Hôpital's Rule. The result can be any number, infinity, or may not exist. |
| Division by zero | Undefined | Not infinity. A distinct and unrelated concept to limits approaching infinity. |
Therefore, while dividing a constant by an increasingly large number approaches zero, division involving infinity in indeterminate forms requires careful evaluation and does not always result in zero. Direct "division by infinity" as a singular arithmetical operation is not defined.
