Choosing a limit procedure AB & BC

Every limit in this unit yields to the same opening move, followed by a branch:

1. Substitute. Always. It is fast and it classifies the problem.
2. A number? That is the limit. Stop.
3. $\frac{0}{0}$? Rewrite: factor, use a conjugate, or clear a complex fraction. Then substitute again.
4. $\frac{k}{0}$ with $k \ne 0$? Unbounded. Check each side's sign and report that the limit does not exist.
5. A piecewise or absolute-value join? Compute both one-sided limits before anything else.
6. Trapped between two bounds? Squeeze Theorem, Topic 1.8.

The branch is decided by evidence rather than by pattern-matching the printed expression, which is why substitution comes first even when the answer looks obvious.

Once you know it is $\frac{0}{0}$, the structure names the tool:

1. Polynomials on top and bottom → factor. Difference of squares and trinomials cover most cases.
2. A square root in a sum or difference → multiply by the conjugate.
3. Fractions inside a fraction → combine over a common denominator first.
4. Absolute values → split into cases by sign, then take one-sided limits.
5. A trigonometric ratio at 0 → look for $\frac{\sin x}{x} \to 1$ or $\frac{1-\cos x}{x} \to 0$.

These are not competing methods. Each is the response to a specific obstruction, and the obstruction is visible once the form is known.

The limit laws are the most over-applied tool in the unit. Before splitting a quotient, confirm the denominator's limit is nonzero. Before splitting a sum, confirm each piece has a limit.

When a piece fails, that is information rather than a dead end. $\lim_{x\to 0} x\sin\left(\frac{1}{x}\right)$ cannot be split, because $\sin\left(\frac{1}{x}\right)$ has no limit at 0. The product still has a limit of 0, found by squeezing. The failure of one tool points to the next.

Three habits save real time.

Substitute before simplifying. Most limits are determinate and finish in one step. Factoring first turns a five-second problem into a one-minute one.

Simplify before deciding a limit fails. A $\frac{0}{0}$ is not a verdict, and most of them resolve to a finite value.

Sketch when the algebra stalls. A rough graph settles one-sided behavior at an asymptote faster than sign analysis, and it catches the case where the two sides run opposite ways.