The Idea of a Limit
Approaching a point: is it the value, or the direction?
Summary
A limit asks which value a function heads toward as it approaches a certain point. The point itself does not have to be one where the function is defined: only its "surroundings" matter. If approaching from the left and from the right leads to the same value, the limit exists.
In depth
The limit is the hidden hero of modern calculus. Both the derivative (f'(x) = lim h→0 [f(x+h)−f(x)]/h) and the integral (the limit of a Riemann sum) are defined using limits. The idea seems intuitive, but its rigorous (epsilon-delta) definition was given in the 19th century by Cauchy and Weierstrass: "for every ε > 0 there is a δ > 0 such that when |x − a| < δ, |f(x) − L| < ε". In plain words: choose x close enough to a and f(x) will be as close to L as you like. An important case: a function can have a limit at a point even if it is undefined there. For example, sin(x)/x is undefined at x = 0 but its limit is 1, so the graph shows a "hole" at that point, yet the value is clear once you look at its neighbourhood.
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