Topic 1.7 Selecting Procedures for Determining Limits: choose an efficient strategy for a given limit, recognizing which technique fits the form of the function.

What this topic is asking

The College Board (Topic 1.7) is a strategy topic: given any limit, decide quickly which method is most efficient. There is nothing new to learn here, only a routine for picking among substitution, algebraic manipulation, the special trigonometric limits, the squeeze theorem, and numerical or graphical estimation.

The decision routine

Matching the tool to the form

• Polynomial or rational, continuous at $a$: direct substitution.
• Rational giving $\frac{0}{0}$: factor numerator and denominator, cancel the $(x - a)$ factor.
• Square root giving $\frac{0}{0}$: multiply by the conjugate.
• Complex (stacked) fraction: combine over a common denominator, then simplify.
• Trigonometric $\frac{0}{0}$: rewrite to reveal $\frac{\sin u}{u} \to 1$ or $\frac{1 - \cos u}{u} \to 0$.
• Trapped between two functions, or hard trig: the squeeze theorem.
• No clean algebra, calculator allowed: a numerical table or a graph for an estimate.

Why "efficient" matters

On the no-calculator section, an exact algebraic method is required, and picking the right one saves time. On the calculator section, a table or graph is fine for an estimate, but an exact method is still faster and exact when it applies. The exam rewards recognizing the form at a glance.

Calculator versus no-calculator strategy

Which procedure is "best" depends partly on which section you are in. On the no-calculator parts, you must produce an exact value with shown algebra, so substitution and the algebraic techniques are the only acceptable routes; a table or graph cannot justify an exact answer there. On the calculator parts, a numerical table or a graph is a legitimate way to estimate a limit, and it is sometimes faster than wrestling with messy algebra - but an exact algebraic method, when it applies cleanly, is still both quicker and exact. The mature approach is to attempt substitution first regardless of section, reach for algebra when you hit $\frac{0}{0}$, and only fall back to numerical or graphical estimation when no clean exact method presents itself and the calculator is allowed. Matching the tool to both the form of the function and the rules of the section is what "selecting procedures" really means.

A worked classification

Run three quick limits through the routine. $\lim_{x \to 2}(x^3 - 1)$: substitution gives $7$, done. $\lim_{x \to 1}\frac{x^2 - 1}{x - 1}$: $\frac{0}{0}$, factor to $x + 1 \to 2$. $\lim_{x \to 0}\frac{\tan x}{x}$: $\frac{0}{0}$, rewrite $\frac{\sin x}{x}\cdot\frac{1}{\cos x} \to 1 \cdot 1 = 1$. The routine handles all three the same way: substitute, classify, apply the matching tool. Notice that the classification step - reading what substitution produced - is what tells you which tool to reach for, so it is never wasted effort even when the limit turns out to need more work.