Limit

A function $f(z)$ is said to have a limit $\lim_{z\to a} f(z)=c$ if, for all $\epsilon>0$, there exists a $\delta>0$ such that $\vert f(z)-c\vert<\epsilon$ whenever $0<\vert z-a\vert<\delta$.

A Lower Limit

$$\mathop{\rm lower} \lim_{n\to \infty} S_n = \underline{\lim_{n\to \infty}} S_n =h$$

is said to exist if, for every $\epsilon>0$, $\vert S_n-h\vert < \epsilon$ for infinitely many values of $n$ and if no number less than $h$ has this property.

An Upper Limit

$$\mathop{\rm upper} \lim_{n\to\infty} S_n = \overline{\lim_{n\to\infty}} S_n=k$$

is said to exist if, for every $\epsilon>0$, $\vert S_n-k\vert < \epsilon$ for infinitely many values of $n$ and if no number larger than $k$ has this property.

Indeterminate limit forms of types $\infty/\infty$ and $0/0$ can be computed with L'Hospital's Rule. Types $0\cdot\infty$ can be converted to the form $0i/0$ by writing

$$f(x)g(x)={f(x)\over 1/g(x)}.$$

Types $0^0$, $\infty^0$, and $1^\infty$ are treated by introducing a dependent variable $y = f(x)g(x)$, then calculating lim $\ln y$. The original limit then equals $e^{\lim\ln y}$.

See also Central Limit Theorem, Continuous, Discontinuity, L'Hospital's Rule, Lower Limit, Upper Limit

References

Courant, R. and Robbins, H. ``Limits. Infinite Geometrical Series.'' §2.2.3 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 63-66, 1996.