Limits and continuity

Share a cake among x people: each one receives 1/x of the cake. For 10 people, one tenth. For 1000 people, one thousandth. As x grows, each share gets closer to 0 without ever reaching it. We say the limit of 1/x as x tends to plus infinity is 0, and we write it with an arrow: as x → +∞, 1/x → 0.

A limit describes the behaviour of a function at its frontiers: when x becomes huge (towards +∞ or -∞), or when x approaches a forbidden value. The result can be a finite number, or +∞ or -∞ if the function shoots off. Example of a forbidden value: 1/x as x approaches 0 from the positive side. Dividing 1 by a tiny number gives a gigantic result: the limit is +∞.

Graphically, a finite limit at infinity shows up as a horizontal asymptote: the curve hugs a horizontal line without ever quite touching it. If the limit of f(x) as x → +∞ is 3, the curve runs along the line y = 3. An infinite limit at a point gives a vertical asymptote, like the line x = 0 for 1/x.

For polynomials, a useful reflex: at infinity, the highest degree term is in charge. So 5 - x², as x → +∞, behaves like -x² and tends to -∞: the -x² crushes the 5.

Continuity happens at one precise point: f is continuous at a if the limit of f(x) as x tends to a exists and equals exactly f(a). Intuitively, the curve can be drawn without lifting the pencil: no jump, no hole. If the limit and the value f(a) do not match, the curve jumps and continuity breaks.