No, 1/0 is undefined in standard arithmetic. However, the limit of 1/x as x approaches 0 can approach infinity, depending on the direction from which x approaches 0.
Here's a breakdown:
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1/0 in Basic Arithmetic: Division by zero is undefined. There is no number that, when multiplied by zero, gives you one.
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Limits and Infinity: Limits describe the behavior of a function as its input approaches a certain value. We are not actually evaluating the function at that value, but rather getting arbitrarily close.
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One-Sided Limits:
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Limit as x approaches 0 from the right (positive side): lim (x→0⁺) 1/x = +∞. As x gets smaller and smaller positive numbers (e.g., 0.1, 0.01, 0.001), 1/x gets larger and larger, approaching positive infinity.
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Limit as x approaches 0 from the left (negative side): lim (x→0⁻) 1/x = -∞. As x gets smaller and smaller negative numbers (e.g., -0.1, -0.01, -0.001), 1/x gets larger and larger in the negative direction, approaching negative infinity.
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The Limit Exists Only If: For the limit of 1/x as x approaches 0 to exist, both the limit from the right and the limit from the left must exist and be equal. Since they are not equal (+∞ vs -∞), the general limit lim (x→0) 1/x does not exist. We often say it is "undefined," or that it "diverges."
In summary:
While the expression 1/0 itself is undefined, in the context of limits, 1/x can approach positive or negative infinity as x approaches 0 from the right or left, respectively. Because these limits are not equal, the limit of 1/x as x approaches 0 does not exist.
