Estimating Limit Values from Tables

A table of values works because a limit is about trend, not about the exact point itself. If you pick x-values marching toward 2, like 1.9, 1.99, 1.999 from the left and 2.1, 2.01, 2.001 from the right, and the corresponding f(x) values keep closing in on some number, that number is the limit's best estimate. You never need to know what happens exactly at x = 2, which matters because the function might not even be defined there.

The precision of your guess is limited by how close your x-values get and how many matching decimal digits the outputs share. If f(1.999) = 3.00000001 and f(2.001) = 2.99999999, you can confidently say the limit looks like 3, because both sides are agreeing to many decimal places. If the outputs only agree to one decimal place, your estimate is much shakier, so you should push the x-values closer before committing to an answer.

This table method sits right next to reading a limit off a graph: both are about watching behavior near a point rather than at it, and both are estimation tools, not proof. A graph shows the trend visually as a curve approaching a height; a table shows the same trend numerically. Neither one guarantees the limit exists or equals the suspected value, since a function can misbehave between the sampled points. That gap is exactly why later tools, like direct substitution, factoring, and limit laws, exist: they confirm algebraically what the table only suggests.