We write:limx → c f(x) = Lorf(x) → Lasx → c

if the functional value f(x) is close to the single real number *L* whenever *x* is close, but not equal, to *c* (on either side of *c*.)

Note that:

- The existence of a limit at
*c*has no relation to the value of the function at*c*. *c*does not even need to be in the domain of*f*.- The function must be defined on both sides of
*c*.

Drag the slider to see the value of the function f(x) = x + 1 as *x* is close to 1.

We write:limx → c^{-} f(x) = K

and call *K* the **limit from the left** or the **left-hand limit** if *f(x)* is close to *K* whenever *x* is close to, but to the left of, *c* on the real number line.

We write:limx → c^{+} f(x) = L

and call *L* the **limit from the right** or the **right-hand limit** if *f(x)* is close to *L* whenever *x* is close to, but to the right of, *c* on the real number line.

For a (two-sided) limit to exist, the limit from the left and the limit from the right must exist and be equal. That is,

limx → c f(x) = Lif and only iflimx → c^{-} f(x) = limx → c^{+} f(x) = L

Drag the slider to see how the value of the function *f(x)* changes with *x*.

Let *f* and *g* be two functions, and assume that:

limx → c f(x) = Landlimx → c g(x) = M

where *L* and *M* are real numbers (both limits exist). Then:

limx → ck = k | limx → cx = x |

limx → c[f(x) + g(x)] = limx → cf(x) + limx → cg(x) = L + M | limx → c[f(x) - g(x)] = limx → cf(x) - limx → cg(x) = L - M |

limx → ck f(x) = klimx → cf(x) = kL | limx → c[f(x) ⋅ g(x)] = [limx → cf(x)] [limx → cg(x)] = LM |

limx → c[f(x)/g(x)] = [limx → cf(x)]/[limx → cg(x)] = L/M if M ≠ 0 | limx → c^{n}√ f(x) = ^{n}√ limx → c f(x) = ^{n}√ L, L > 0 for n even |

- limx → cf(x) = f(c) for
*f*any polynomial function. - limx → cr(x) = r(c) for
*r*any rational function with a nonzero denominator at x = c.

- If limx → cf(x) = 0 and limx → cg(x) = 0,
then limx → cf(x)/g(x) is said to be

**indeterminate**, or, more specifically, a**0/0 indeterminate form**. - The limit of an indeterminate form may or may not exist.

If limx → cf(x) = L, L ≠ 0 , and limx → cg(x) = 0

then limx → cf(x)/g(x) does not exist.