Limit of a function

${\displaystyle x}$	${\displaystyle {\frac {\sin x}{x}}}$
1	0.841471...
0.1	0.998334...
0.01	0.999983...

Although the function ⁠${\displaystyle {\tfrac {\sin x}{x}}}$⁠ is not defined at zero, as x becomes closer and closer to zero, ⁠${\displaystyle {\tfrac {\sin x}{x}}}$⁠ becomes arbitrarily close to 1. In other words, the limit of ⁠${\displaystyle {\tfrac {\sin x}{x}},}$⁠ as x approaches zero, equals 1.

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In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function.

Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, the output value can be made arbitrarily close to L if the input to f is taken sufficiently close to p. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, then we say the limit does not exist.

The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.