Limits and Continuity — AP Calculus AB
1. Limit Definitions and Notation ★★☆☆☆ ⏱ 3 min
The intuitive definition of a limit, most heavily tested on AP Calculus AB, states that $\lim_{x \to a} f(x) = L$ means $f(x)$ gets arbitrarily close to $L$ as $x$ approaches $a$ from both sides, without ever needing to equal $a$ at the point itself.
- **Left-hand limit**: $\lim_{x \to a^-} f(x)$: value approached by $f(x)$ as $x$ approaches $a$ from values smaller than $a$
- **Right-hand limit**: $\lim_{x \to a^+} f(x)$: value approached by $f(x)$ as $x$ approaches $a$ from values larger than $a$
A core rule: $\lim_{x \to a} f(x)$ exists **if and only if** the left-hand and right-hand limits are equal. Formal epsilon-delta proofs are rarely tested on AB, but you should recognize the definition for multiple-choice questions.
2. Computing Limits Algebraically and Graphically ★★☆☆☆ ⏱ 5 min
You will be asked to compute limits from both graphs and algebraic expressions on the exam. For graphical computation, follow these steps:
- Trace the graph from the left of $x=a$ to identify the left-hand y-value approached
- Trace the graph from the right of $x=a$ to identify the right-hand limit
- If the two values match, that is the two-sided limit: ignore the actual value of $f(a)$ even if there is a hole, jump, or isolated point at $x=a$
For algebraic computation, always start with direct substitution: if $f(x)$ is continuous at $a$ (polynomials, rational functions with non-zero denominators, trig, exponential, and log functions in their domains), substitute $x=a$ directly to get the limit. If you get the indeterminate form $\frac{0}{0}$, use factoring or rationalization to cancel problematic terms.
3. Indeterminate Forms and L'Hôpital's Rule ★★★☆☆ ⏱ 3 min
Indeterminate forms are expressions that do not yield a clear limit when you substitute directly. The two most common forms tested on AP Calculus AB are $\frac{0}{0}$ and $\frac{\pm\infty}{\pm\infty}$.
Critical exam rules: You cannot use L'Hôpital's for non-indeterminate forms, and you must differentiate numerator and denominator separately (do not use the quotient rule). You can apply it multiple times if the new limit is still indeterminate.
4. Continuity and the Intermediate Value Theorem ★★★☆☆ ⏱ 4 min
A function is continuous at a point if the limit approached at the point matches the actual function value. Three conditions must all be satisfied for continuity at $x=a$:
- $f(a)$ is defined (the function exists at $x=a$)
- $\lim_{x \to a} f(x)$ exists (left and right limits are equal)
- $\lim_{x \to a} f(x) = f(a)$ (the limit equals the function value)
Discontinuities are categorized as removable (hole, limit exists but does not match $f(a)$), jump (left/right limits exist but are unequal), and infinite (one/both one-sided limits are infinite).
5. Asymptotes and End Behaviour ★★★☆☆ ⏱ 3 min
Asymptotes describe function behaviour near discontinuities and end behaviour for very large/small inputs. AP Calculus AB primarily tests vertical and horizontal asymptotes for rational functions:
- **Vertical Asymptote (VA)**: $x=a$ is a VA if $\lim_{x \to a^+} f(x) = \pm\infty$ or $\lim_{x \to a^-} f(x) = \pm\infty$. For rational functions, VAs occur at denominator zeros where the numerator is non-zero.
- **Horizontal Asymptote (HA)**: $y=L$ is a HA if $\lim_{x \to \pm\infty} f(x) = L$, describing end behaviour. For rational functions with numerator degree $n$ and denominator degree $d$:
- - If $n < d$: HA at $y=0$
- - If $n = d$: HA is the ratio of leading coefficients
- - If $n > d$: No horizontal asymptote
Common Pitfalls
Why: Students get used to direct substitution for polynomials and apply it incorrectly to discontinuous or piecewise functions
Why: Students treat L'Hôpital's as a universal limit shortcut without checking required preconditions
Why: Students focus on the sign change and skip the required precondition that is graded explicitly
Why: Students associate all denominator zeros with vertical asymptotes without checking for common factors
Why: Students incorrectly assume left and right limits are always equal