Why is 1∞ an indeterminate form while 0∞=0? If 0⋅0⋅0⋯=0 shouldn't 1⋅1⋅1⋯=1?
Answer
To say that 1∞ is an indeterminate form means that there is more than one object that can be lim where f(x)\to1 and g(x)\to\infty, so that the limit depends on which functions f and g are.
Thus
\left. \begin{align} & \lim_{x\to\infty} \left(1+\frac 1 x\right) = 1 \quad\text{and} \quad \lim_{x\to\infty} \left( 1 + \frac 1 x \right)^x = e \\[10pt] & \qquad \text{and} \\[10pt] & \lim_{x\to\infty} \left( 1 - \frac 1 x\right) = 1 \quad \text{and} \quad \lim_{x\to\infty} \left( 1 - \frac 1 x\right)^x = \frac 1 e. \end{align} \right\} \longleftarrow \text{two different numbers}
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