Topic 1.4 Estimating Limit Values from Tables: use a table of values approaching a point from both sides to estimate one-sided and two-sided limits.

What this topic is asking

The College Board (Topic 1.4) wants you to estimate a limit numerically from a table of $x$-values approaching a point from both sides. This matters most when direct substitution gives an indeterminate form like $\frac{0}{0}$, so you cannot just plug in. The calculator section often hands you a table or asks you to build one.

Building and reading a table

Choose $x$-values that approach $a$ from each side, getting progressively closer, for example $a - 0.1, a - 0.01, a - 0.001$ from the left and $a + 0.1, a + 0.01, a + 0.001$ from the right. Evaluate $f$ at each and look for the value the outputs settle toward.

The indeterminate-form case

The reason this topic exists is functions like $f(x) = \frac{x^2 - 9}{x - 3}$, where substitution gives $\frac{0}{0}$. The function is undefined at $x = 3$, yet a table shows the outputs approaching a clean value:

Both sides head to $6$, so $\lim_{x \to 3} f(x) \approx 6$. (Algebraically, $\frac{x^2-9}{x-3} = x + 3 \to 6$, confirming the estimate.)

Choosing good input values

The quality of a table estimate depends entirely on the inputs you choose. They should approach $a$ from both sides and should get progressively closer, ideally by factors of ten ($a \pm 0.1$, $a \pm 0.01$, $a \pm 0.001$), so that you can watch the digits of the limit stabilize. If you only sample one side, you have a one-sided estimate, not a two-sided limit. If your values are not close enough to $a$, the trend may not have settled and you will read off the wrong number. On the calculator section, this is exactly what a graphing calculator's table feature is for: set the table to step in small increments around $a$ and read the convergence directly. Reporting the limit to as many digits as have clearly stabilized, while noting that a table gives an estimate rather than a proof, is the honest and full-credit way to present a numerical limit.

When the table says "does not exist"

If the left column heads toward one number and the right column toward another, report that the two-sided limit does not exist. Also be alert to outputs that grow without bound (for example $100, 10{,}000, 1{,}000{,}000$): that signals an infinite limit, so the finite limit does not exist. A subtler trap is a function that oscillates, where the tabulated values jump around without settling no matter how close you get; that too means no limit exists, and a coarse table can hide it, which is one reason an exact algebraic method is preferred whenever it is available. Treat the table as a tool for estimation and confirmation, not as the final word when an exact technique applies.