Selecting procedures for determining limits — AP Calculus BC
1. Direct Substitution for Determinate Limits ★☆☆☆☆ ⏱ 3 min
Direct substitution is always the first procedure you should test, as it is the fastest and simplest method. It works when a function $f(x)$ is continuous at $x=a$, the point we are approaching. If you get a finite real number after substitution, that is your limit and you are done. If you get $c/0$ for $c \neq 0$, the limit is infinite or does not exist, which is still a determinate result.
Exam tip: Always test direct substitution first. Roughly 70% of basic AP limit problems can be solved this way, saving valuable exam time.
2. Algebraic Manipulation for Indeterminate $0/0$ Limits ★★☆☆☆ ⏱ 4 min
When direct substitution gives an indeterminate $0/0$ form, algebraic manipulation is the next procedure to try for problems with polynomials or radicals. If both numerator and denominator are zero at $x=a$, $(x-a)$ is always a common factor that can be canceled, which does not change the limit since the limit only depends on values near $x=a$, not at $x=a$.
Exam tip: After canceling a common factor, always re-test direct substitution on the simplified expression. You will almost always get a finite answer after simplification for $0/0$ polynomial/radical limits.
3. L'Hospital's Rule for Indeterminate Forms ★★★☆☆ ⏱ 4 min
L'Hospital's Rule is the go-to procedure for indeterminate forms when algebraic manipulation is not feasible, such as for problems involving transcendental functions (trigonometric, exponential, logarithmic) or hard-to-factor high-degree polynomials. It only applies to indeterminate $0/0$ or $\infty/\infty$ forms, but can be adapted for other indeterminate forms by rewriting them as a fraction.
Exam tip: Never apply L'Hospital's Rule to determinate forms, and never differentiate the entire quotient with the quotient rule—you must differentiate numerator and denominator separately.
4. Squeeze Theorem for Oscillating Bounded Limits ★★★☆☆ ⏱ 3 min
The Squeeze Theorem (also called the Sandwich Theorem) is the correct procedure for limits involving bounded oscillating functions, like $\sin\left(\frac{1}{x}\right)$ or $\cos\left(\frac{1}{x}\right)$, multiplied by a function that approaches zero (or for bounded functions divided by a function approaching infinity). Trigonometric functions like sine and cosine are always bounded between $-1$ and $1$, which lets us squeeze the function between two bounds that approach the same limit.
Exam tip: If you see a trigonometric function with $\frac{1}{x}$ or another term that causes oscillation as $x \to 0$ or $x \to \infty$, the Squeeze Theorem is almost always correct—algebra and L'Hospital's Rule will not work here.
Common Pitfalls
Why: You jumped to L'Hospital's Rule without checking the form after direct substitution; the original limit gives $2/0$, a determinate infinite form, not indeterminate.
Why: You confuse the value of the function at $x=2$ with the limit as $x \to 2$, leading to incorrect conclusions about continuity.
Why: You focus on differentiating the more complex numerator and overlook the simple denominator.
Why: You default to algebraic manipulation for any $0/0$ limit, regardless of function type.
Why: You forget the Squeeze Theorem applies to limits at infinity as well as finite points.