Estimating Limit Values from Tables

A limit asks what f(x) is heading toward as x gets close to some number c, not what happens exactly at c. A table turns that question into something you can see: pick x-values that creep toward c, compute f(x) for each, and watch the outputs. If they settle down near one number, that number is your estimate for the limit. This matters because plenty of functions are undefined or weird exactly at c, yet behave perfectly predictably right around it.

The key move is approaching from both sides. One-sided creeping isn't enough, since a function can head toward different values from the left and the right. So build two columns: x-values below c getting closer (like 1.9, 1.99, 1.999 for c = 2) and x-values above c getting closer (2.1, 2.01, 2.001). Compute f(x) for each. If both sides crowd toward the same number, that shared value is the limit estimate. If they disagree, the limit doesn't exist at that point.

This table method is a stepping stone. It builds the intuition that a limit is a trend, not a plug-in value, which is exactly the idea that algebraic limit laws and later, more exact techniques will formalize. Tables are especially useful for the special trig limits, sin(x)/x and (1 - cos(x))/x², where direct substitution gives 0/0 and tells you nothing, but a table reveals the hidden finite value immediately.