Limits from a table AB & BC

To estimate $\lim_{x\to a} f(x)$, choose inputs that close in on $a$ from both directions and never equal it:

$$a - 0.1,\ a - 0.01,\ a - 0.001 \qquad\text{and}\qquad a + 0.001,\ a + 0.01,\ a + 0.1.$$

Read down each side and ask what the outputs are settling toward. If both sides settle on the same value, that is your estimate. If they settle on different values, the limit does not exist, and the table has told you so.

A table that only descends from above establishes at most a right-hand limit. That is a real result, but it is not the two-sided limit unless the other side is checked.

Convergence looks like digits locking into place from the left of the number. Estimating $\lim_{x\to 2}\frac{x^2-4}{x-2}$:

1. $x = 1.9 \Rightarrow 3.9$
2. $x = 1.99 \Rightarrow 3.99$
3. $x = 1.999 \Rightarrow 3.999$
4. $x = 2.001 \Rightarrow 4.001$
5. $x = 2.01 \Rightarrow 4.01$

Both sides are pinning down $4$. Note that $x = 2$ itself gives $\frac{0}{0}$, so the function is undefined exactly where the limit lives. That is normal and is the whole reason to build a table.

Contrast a table whose outputs read $3.7, 3.9, 4.4, 5.1$: those are moving, not settling, and support no conclusion yet.

When substitution gives $\frac{0}{0}$, that is an indeterminate form. It does not mean the limit is 0, or 1, or that it fails to exist. It means the form alone carries no information and the question is still open.

Three different functions all give $\frac{0}{0}$ at $x = 0$ and have three different limits:

1. $\frac{x}{x} \to 1$
2. $\frac{x^2}{x} \to 0$
3. $\frac{x}{x^2} \to$ unbounded, so no limit

Since the form is identical in all three, the form cannot be what decides. A table resolves them instantly, and algebra confirms it. The same applies to $\frac{\infty}{\infty}$, $0\cdot\infty$, and $1^{\infty}$: each is a prompt to do more work.

Numerical evidence is suggestive, never conclusive. A table samples finitely many inputs, and a function can misbehave between them or closer in than you sampled. $\sin\left(\frac{\pi}{x}\right)$ evaluated at $x = 1, 0.1, 0.01$ returns $0$ every time, suggesting a limit of 0 at $x=0$; the true behavior oscillates forever and has no limit. The table was not wrong, it was unlucky.

So use tables to form a conjecture and to check an algebraic answer for plausibility, and use algebra to settle it. On the exam, a question asking you to estimate from a table wants the estimate; a question asking you to evaluate wants the algebra.