AP Calculus AB: a complete guide to evaluating limits algebraically on the exam

What evaluating limits actually demands

Evaluating limits is the gateway skill of AP Calculus AB, and the no-calculator section leans on it heavily. The College Board wants you to find an exact value, show the algebra that justifies it, and recognize instantly which technique a given limit needs. This guide ties together the matching dot-point pages, each with its own practice: limits using algebraic properties, limits using algebraic manipulation, selecting procedures for limits, the squeeze theorem, infinite limits and vertical asymptotes, and limits at infinity and horizontal asymptotes.

The substitution-first routine

Every limit starts the same way: substitute the target value. The result tells you the problem type.

• A real number means the function is continuous at $a$, and that number is the limit. Done.
• A nonzero number over zero ($\frac{k}{0}$, $k \neq 0$) means an infinite limit; the finite limit does not exist, and you do sign analysis to report $+\infty$ or $-\infty$.
• $\frac{0}{0}$ is indeterminate: the limit may exist, but you must rewrite the function first.

This single routine prevents the most common mistake of all, which is reporting $\frac{0}{0}$ as if it were a value.

Resolving 0/0 by rewriting

When substitution gives $\frac{0}{0}$, the numerator and denominator share a hidden common factor. Match the obstacle to the tool.

Factor and cancel

For rational expressions, factor both parts and cancel the factor that vanishes at $a$:

Multiply by the conjugate

When a square root creates the $\frac{0}{0}$, multiply top and bottom by the conjugate to convert the root into a difference of squares:

Combine a complex fraction

When fractions are stacked, combine them over a common denominator before cancelling:

The special trigonometric limits

Two limits (in radians) resolve most trigonometric indeterminate forms:

The trick is to rewrite until one pattern appears. For $\lim_{x \to 0} \frac{\sin 3x}{x}$, multiply and divide to match $\frac{\sin 3x}{3x}$:

Infinite limits and sign analysis

When substitution gives $\frac{k}{0}$ with $k \neq 0$, the line $x = a$ is a vertical asymptote and the limit is infinite. Determine the sign on each side: check the sign of the numerator near $a$ and the sign of the (tiny) denominator just left and just right of $a$. Matching signs give $+\infty$; opposite signs give $-\infty$. A squared factor in the denominator gives the same infinity on both sides.

Limits at infinity

For end behavior as $x \to \pm\infty$, divide every term by the highest power of $x$ in the denominator and send the $\frac{1}{x^n}$ terms to zero:

The degree rule summarizes the outcome: bottom-heavy gives $0$, equal degrees give the leading-coefficient ratio, and top-heavy gives $\pm\infty$. Watch the sign of $\sqrt{x^2} = |x|$, which equals $-x$ as $x \to -\infty$ and can produce two different horizontal asymptotes.

How limit evaluation is examined

• Direct evaluation. Substitute and simplify; report exact values, not decimals, on the no-calculator part.
• Indeterminate forms. Recognize $\frac{0}{0}$ and choose factoring, a conjugate, fraction-combining, or a special trig limit.
• Infinite behavior. Identify vertical asymptotes with sign analysis and horizontal asymptotes with the degree rule.
• Connecting representations. Confirm an algebraic limit against a graph or table when the question gives one.

Check your knowledge

A mix of direct, indeterminate, infinite and at-infinity limits. Work them under no-calculator conditions, then check against the solutions.

1. Evaluate $\lim_{x \to 1}(2x^2 - 3x + 5)$. (1 mark)
2. Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$. (2 marks)
3. Evaluate $\lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x}$. (2 marks)
4. Evaluate $\lim_{x \to 0} \frac{\sin 4x}{x}$. (2 marks)
5. Evaluate $\lim_{x \to 2^+} \frac{1}{x - 2}$ and state its sign. (2 marks)
6. Evaluate $\lim_{x \to \infty} \frac{5x^2 + 2}{3x^2 - x}$. (2 marks)
7. Evaluate $\lim_{x \to -\infty} \frac{2x + 1}{\sqrt{x^2 + 1}}$. (2 marks)
8. Evaluate $\lim_{x \to 4} \frac{\frac{1}{x} - \frac{1}{4}}{x - 4}$. (2 marks)