One-sided and infinite limits

Just left of 2, |x − 2|/(x − 2) is always −1; just right of 2 it is always 1. The two sides never agree.

Limits from the left and from the right

Sometimes the two sides of a point tell different stories, and it helps to keep them apart. The limit from the left, written lim_x→a⁻ f(x), uses only x-values less than a. The limit from the right, lim_x→a⁺ f(x), uses only x-values greater than a. For g(x) = |x − 2|/(x − 2), any x less than 2 makes the top equal to −(x − 2), so g(x) = −1; any x greater than 2 gives g(x) = 1. So the left limit is −1 and the right limit is 1.

The connection to the ordinary two-sided limit is simple and worth memorizing: lim_x→a f(x) = L exactly when both one-sided limits exist and both equal L. For g the sides disagree, so lim_x→2 g(x) does not exist, often abbreviated DNE. OpenStax also uses a piecewise example: f(x) = x + 1 for x < 2 and f(x) = x² − 4 for x ≥ 2. From the left the values approach 2 + 1 = 3; from the right they approach 4 − 4 = 0. The graph jumps, and the two-sided limit does not exist.

When there is no number at all

A limit can fail to exist in a second way. Look at sin(1/x) near 0. As x shrinks, 1/x grows without bound, and the sine of a fast-growing number keeps swinging between −1 and 1. OpenStax picks x-values such as 2/π, 2/(3π), 2/(5π) and so on, where the function alternates 1, −1, 1, −1 forever, getting closer to 0 all the time. The values never settle on one number from either side, so neither one-sided limit exists.

Infinite limits and vertical asymptotes

The third kind of behaviour is growth without bound. For h(x) = 1/(x − 2)², the outputs at x = 2.01 and 1.99 are both 10,000, and at 2.001 and 1.999 they are 1,000,000. They do not approach a number; they grow past any number you name. We write lim_x→2 1/(x − 2)² = +∞. This is a description of how the limit fails, not a claim that infinity is a number. For 1/x near 0 the two sides go in opposite directions: −∞ from the left and +∞ from the right.

Whenever a one-sided limit at a is +∞ or −∞, the vertical line x = a is a vertical asymptote: the graph runs up or down alongside it, closer and closer. Rational functions often have one where the denominator is zero and the numerator is not. OpenStax's example 1/(x + 3)⁴ has a vertical asymptote at x = −3. Be careful with zero over zero: (x² − 4)/(x − 2) has a zero denominator at 2 but a hole there, not an asymptote, because the zero cancels.