Limit of a function

Limit of a function at a point

Suppose f : U -> R is a function, where U is a subset of the real numbers. If p and L are two real numbers, we say that the limit of f(x) as x approaches p is L and write

$\lim _{x\to p}f(x)=L$

if and only if

for every ε > 0 there exists a δ > 0 such that for all x in U with 0 < |x - p| < δ, we have |f(x) - L| < ε.

This is equivalent to saying

for every convergent sequence (x_n) in U - {p} with limit equal to p, the sequence (f(x_n)) converges with limit L.

Note that the function does not have to be defined at p, and in any case, the value f(p) is irrelevant for the determination of the limit of f at p.

We also consider situations where either p or L or both are positive or negative infinity. We say that f(x) approaches positive infinity (+∞) as x approaches p if and only if

for every R > 0 there exists a δ > 0 such that whenever 0 < |x - p| < δ then f(x) > R.

We say that the limit of f(x) as x approaches positive infinity is L if and only if

for every ε > 0 there exists an S > 0 such that whenever x > S, then |f(x) - L| < ε.

Finally, we say that the limit of f(x) is positive infinity as x approaches positive infinity if and only if

for every R > 0 there exists an S > 0 such that whenever x > R, then f(x) > S.

The definitions for negative infinity are analogous.

Occasionally, it is useful to approach the point p only from one side. The one-sided limit of f(x) as x approaches p from the right is L, written as

$\lim _{x\to p+}f(x)=L$

if and only if

for every ε > 0 there exists a δ > 0 such that for all x in U with 0 < x - p < δ, we have |f(x) - L| < ε.

Left-sided limits are obtained by replacing x - p in the last definition by p - x. By replacing ε by S as above, we can also define one-sided limits that are infinite.

Examples

• The limit of 1/x as x approaches infinity is 0.
• The two-sided limit of 1/x as x approaches 0 does not exist. The limit of 1/x as x approaches 0 from the right is +∞.
• The limit of x² as x approaches 3 of is 9. (In this case the value of the function happens to be well defined at the point, and the function's value is the same as its limit.)
• The limit of x^x as x approaches 0 is 1.
• The limit of ((a + x)² - a² ) / x as x approaches 0 is 2a.
• The one-sided limit of sqrt(x²)/x as x approaches 0 from the right is 1; the one-sided limit from the left is -1.
• The limit of x sin(1/x) as x approaches positive infinity is 1.
• The limit of (cos(x) - 1)/x as x approaches 0 is 0.

Properties

If the limit of f(x) as x approaches p exists (which need not be the case), and if there exists at least one sequence (x_n) with elements in U - {p} and limit equal to p, then the limit of f(x) as x approaches p is uniquely determined by f and p.

The two-sided limit of f(x) as x approaches p exists if and only if the left-sided and right-sided limits exist and are equal.

The function f is continuous at the point p if and only if the two-sided limit of f(x) as x approaches p is finite and equal to f(p).

Taking the limit of functions is compatible with the algebraic operations: If

$\lim _{x\to p}f_{1}(x)=L_{1}$

and

$\lim _{x\to p}f_{2}(x)=L_{2}$

then

$\lim _{x\to p}(f_{1}(x)+f_{2}(x))=L_{1}+L_{2}$

and

$\lim _{x\to p}(f_{1}(x)\times f_{2}(x))=L_{1}\times L_{2}$

and

$\lim _{x\to p}\left({\frac {f_{1}(x)}{f_{2}(x)}}\right)={\frac {L_{1}}{L_{2}}}$

(the latter provided that f₂(x) is non-zero in a neighborhood of p and L₂ is non-zero as well).

These rules are also valid for one-sided limits, for the case p = ±∞, and also for infinite limits using the rules

• q + ∞ = ∞ for q ≠ -∞
• q × ∞ = ∞ if q > 0
• q × ∞ = -∞ if q < 0
• q / ∞ = 0 if q ≠ ± ∞

(see extended real number line).

Note that there is no general rule for the case q / 0; it all depends on the way 0 is approached. Some cases, for instance 0/0, 0×∞ ∞-∞ or ∞/∞, are also not covered by these rules but the corresponding limits can usually be determined with l'Hôpital's rule.