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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.