List of limits

This is a list of limits for common functions such as elementary functions. In this article, the terms a, b and c are constants with respect to x.

${\displaystyle \lim _{x\to c}f(x)=L}$ if and only if ${\displaystyle \forall \varepsilon >0\ \exists \delta >0:0<|x-c|<\delta \implies |f(x)-L|<\varepsilon }$. This is the (ε, δ)-definition of limit.

The limit superior and limit inferior of a sequence are defined as ${\displaystyle \limsup _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\sup _{m\geq n}x_{m}\right)}$ and ${\displaystyle \liminf _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\inf _{m\geq n}x_{m}\right)}$.

A function, ${\displaystyle f(x)}$, is said to be continuous at a point, c, if $${\displaystyle \lim _{x\to c}f(x)=f(c).}$$

If ${\displaystyle \lim _{x\to c}f(x)=L}$ then:

In general, if g(x) is continuous at L and ${\displaystyle \lim _{x\to c}f(x)=L}$ then

If ${\displaystyle \lim _{x\to c}f(x)=L_{1}}$ and ${\displaystyle \lim _{x\to c}g(x)=L_{2}}$ then:

In these limits, the infinitesimal change ${\displaystyle h}$ is often denoted ${\displaystyle \Delta x}$ or ${\displaystyle \delta x}$. If ${\displaystyle f(x)}$ is differentiable at ${\displaystyle x}$,