Dividing by zero
2 + 2 = 4 \\
4 -1 = 3
3 . 5 = 3 + 3 + 3 + 3 + 3 \\
5 . 3 = 5 + 5 + 5
\frac{15}{5} = 3 \\
15 -5 -5 -5 = 0
\frac{1}{0} = 1 - \underbrace{ 0 - 0 - 0 - 0 - 0 - 0 - ...}_{ \infty ?} \newline
Examples
\frac{1}{0} = \infty ?
First approach
\frac{1}{1} = 1 \\
\frac{1}{0.1} = 10 \\
\frac{1}{0.01} = 100 \\
\frac{1}{0.001} = 1000 \\
\frac{1}{0.0001} = 10000 \\
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Examples
Second approach
\frac{1}{0} = \infty ?
\frac{3}{1} = 3 \\
\frac{3}{0.1} = 30 \\
\frac{3}{0.01} = 300 \\
\frac{3}{0.001} = 3000 \\
\frac{3}{0.0001} = 30000 \\
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Examples
Second approach
\frac{3}{0} = \infty ?
Weird
\frac{1}{0} = \infty = \frac{3}{0} \\
\frac{1}{0} = \frac{3}{0} \\
1 = 3 ???
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1 \neq 3
This is fundamentally wrong !!!
Undefined
\frac{1}{0} = undefined
Another approach to zero
\frac{1}{-1} = -1 \\
\frac{1}{-0.1} = -10 \\
\frac{1}{-0.01} = -100 \\
\frac{1}{-0.001} = -1000 \\
\frac{1}{-0.0001} = -10000 \\
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\frac{1}{0} = - \infty
There is not number you can pick that make sense !!! Therefore, Undefinable
\lim_{x\to0} \frac{1}{x} = \text{does not exist}
\lim_{x^{+}\to0} \frac{1}{x} = \infty
\lim_{x^{-}\to0} \frac{1}{x} = -\infty
Dividing by zero
By Michal Danco
