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Would it be correct to say |1/0| = ∞?

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Not infinite - undefined.  
There's a difference.   
Dividing by zero just doesn't make sense.
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No, 1/0 is nonsense.
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Do not confuse approaching 0 with dividing by 0 , they are two completely different things .
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The answer here is that infinity is not a number, so you can’t say an expression is equal to it.

A more precise statement would be that

|1/n| -> infinity as n -> 0
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No.  

The limit as n approaches zero of 1/n is infinity, but it’s asymptotic.  1/0 is undefined.  

Sorry.  That was very doctrinaire of me.  Let me try again.   

0 * 5 = 0

0 * 100 = 0

0 * n = 0,  for all n

What do you multiply 0 by to get 1?   That is, how many zeroes do you need to add together to equal 1?  No matter how many you have, the sum will always be zero.

Edit: formatting
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Infinity is a concept, not a number.  "1/0 = infinity" makes as much sense as "1/0 = hungry".

1/0 has no solution. 0/0 has an infinite number of solutions.

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