Concept preview

What you will be able to do

1. 01Build a table of function values at inputs approaching the target from both sides and read the trend.
2. 02Decide the limit (or that it likely does not exist) from the numerical pattern, using both-sided agreement.
3. 03Recognize the cautions of numerical estimation: it suggests but does not prove a limit, and can be fooled by coarse steps.

To estimate $\lim_{x\to c}f(x)$ from a table, choose inputs that step toward $c$ from BOTH sides and shrink the gap: for $c=2$, use $x=1.5,\,1.8,\,1.9,\,1.99$ from the left and $x=2.5,\,2.2,\,2.1,\,2.01$ from the right. Evaluate $f$ at each and look for a single number the outputs are converging to. If the left column and the right column close in on the same value $L$, that is your estimate for the limit.