Determining limits using algebraic manipulation — AP Calculus BC
1. Factoring to Resolve 0/0 Indeterminate Forms ★★☆☆☆ ⏱ 4 min
The most common indeterminate form you will encounter when evaluating limits of rational functions at a finite point is 0/0. This form arises when the numerator and denominator of the rational function share a common root at the point $x=a$ you are approaching. That means both polynomials share a common factor of $(x-a)$, which can be canceled to simplify the function.
Exam tip: Always test if the polynomial has a root at the x-value you are approaching. If $p(a) = 0$, $(x-a)$ is guaranteed to be a factor, so you can use grouping or polynomial division to pull it out quickly.
2. Rationalizing to Resolve Indeterminate Forms with Radicals ★★☆☆☆ ⏱ 3 min
When an indeterminate 0/0 or $0 \cdot \infty$ form includes radicals (square roots, cube roots) in the numerator or denominator, factoring alone cannot resolve the form because the zero term is hidden under the radical. The solution is rationalization: multiplying the numerator and denominator by the conjugate of the radical expression to eliminate the radical and reveal the common zero factor.
The conjugate of a binomial $\sqrt{A} - B$ is $\sqrt{A} + B$. Multiplying these gives a difference of squares: $(\sqrt{A} - B)(\sqrt{A} + B) = A - B^2$, which eliminates the radical entirely. After expanding, you will almost always find a common factor that can be canceled, leaving a simplified expression ready for direct substitution.
Exam tip: Always multiply both the numerator and denominator by the conjugate. Changing only the numerator changes the value of the expression, which leads to an incorrect limit result.
3. Dividing by Highest Power of x for Limits at Infinity ★★★☆☆ ⏱ 4 min
When evaluating limits as $x \to \infty$ or $x \to -\infty$ for rational functions or radical functions, you almost always get the indeterminate form $\infty/\infty$. The core intuition here is that as $x$ becomes very large in magnitude, the highest power term in the expression dominates all lower-power terms, which become negligible.
The standard technique is to divide every term in the numerator and denominator by the highest power of $x$ present in the denominator. Then use the rule that $\lim_{x \to \pm \infty} \frac{1}{x^n} = 0$ for any positive $n$ to eliminate all lower-power terms, leaving a constant limit. This technique is commonly used to find horizontal asymptotes, a frequent AP exam question.
Exam tip: When pulling terms out of a square root for $x \to -\infty$, remember $\sqrt{x^2} = |x| = -x$ for negative $x$. Failing to adjust the sign is the most common error on this type of problem.
4. AP Style Concept Check ★★★☆☆ ⏱ 3 min
Common Pitfalls
Why: Students confuse the value of the limit as $x$ approaches $a$ with the value of the function at $x=a$
Why: Students memorize $\sqrt{x^2}=x$ from algebra and forget the absolute value rule
Why: Students assume dividing by the largest power overall is correct, regardless of where it is located
Why: Students assume there are no more common factors after one cancellation
Why: Students focus on eliminating the radical and forget that changing the numerator changes the value of the expression