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.