Limits, intuitively

Imagine a wall 1 m away and each time you walk half of the remaining distance: first 50 cm, then 25 cm, then 12.5 cm and so on. You never quite reach the wall, but you get arbitrarily close. A limit describes exactly this behavior: the value something heads toward, even if it never actually arrives there.

Approaching a value

Look at the numbers $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ...$, or as decimals 1, 0.5, about 0.333, 0.25, ... They get smaller and smaller and press up against 0. None of the terms is ever exactly 0, but they come as close to 0 as you like. We say: the limit of this sequence is 0.

From both sides

Often we ask where a function heads as $x$ approaches a certain spot. For the spot 2 this is written in short as $x \to 2$, read as: x goes to 2. Approach that spot from the left with smaller values and from the right with larger ones. If both paths land at the same height, that is the limit at this spot, no matter what happens right at the point itself.

Behavior at infinity

You can also ask what happens as $x$ grows without bound, that is, runs to infinity. For $\frac{1}{x}$ the fraction gets tinier and tinier and approaches 0. The graph nestles up against the x-axis without touching it. Lines that a curve approaches ever more closely are called asymptotes.