Indeterminate form

Summary

An indeterminate form is a type of limit where the limits of the parts do not determine the limit of the whole. The seven common forms are $0/0$, $\infty/\infty$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^{\infty}$, and $\infty^0$.

Remarks

For example, $x \to 0$ and $x^2 \to 0$, but $x^2/x \to 0$, $x/x \to 1$, and $x/x^2$ has no finite limit. All three are of the form $0/0$. L'Hôpital's rule treats the forms $0/0$ and $\infty/\infty$. Logarithms turn the forms $0^0$, $1^{\infty}$, and $\infty^0$ into the form $0 \cdot \infty$. Forms such as $1/0^+$ and $0^{\infty}$ are not indeterminate.

Explanations

L1 human

Some limits cannot be found by plugging in the limits of the parts. Then a finer analysis is needed, because many different answers are possible.