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Thought I'd throw together some common failed attempts to create infinities

\[1/0\]

Why it doesn't work: This is an undefined expression, because zero is not in the domain of the second argument of the division function.

\[-\log 0\]

Why it doesn't work: This is an undefined expression, because zero is not in the domain of any logarithm function.

\[\infty \text{ in the extended real numbers}\]

Why it doesn't work: Googology is the process of searching for increasingly large finite real numbers. This is not a real number.

\[\lim_{x \rightarrow 0} 1/x\]

Why it doesn't work: The limit is undefined.

\[\lim_{x \rightarrow 0^+} 1/x\]

Why it doesn't work: The limit is undefined. If we allow use of the extended real numbers, we have the same problem as before.

\[\int_0^\infty 1 dx\]

Why it doesn't work: The limit is undefined.

\[x \text{ where } x = 10^x\]

Why it doesn't work: No real number satisfies this property (although an infinite family of complex numbers does).

\[\text{last member of sequence }1,\,2,\,3,\,4,\,\ldots\]

Why it doesn't work: An infinite sequence by definition has no last member.

\[\delta(0) \text{ (Dirac delta function)}\]

Why it doesn't work: The Dirac delta function is only rigorously valid inside an integral.

\[\zeta(1)\]

Why it doesn't work: It's not in the domain.

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