Calculus AB · Unit 1: Limits and Continuity · 14 min read · Updated 2026-05-10
Estimating limit values from graphs — AP Calculus AB
AP Calculus AB · Unit 1: Limits and Continuity · 14 min read
1. Core Concept of Limits from Graphs★☆☆☆☆⏱ 3 min
Estimating limit values from graphs uses the visual behavior of a function’s curve near $x=a$ to find the output value the function approaches as $x$ gets arbitrarily close to $a$, regardless of the actual value of $f(a)$. This is Topic 1.2 in AP Calculus AB Unit 1, which makes up 10-12% of the total AP exam score, appearing in both multiple-choice and free-response sections.
2. One-Sided and Two-Sided Limits★★☆☆☆⏱ 4 min
A one-sided limit describes the output a function approaches when approaching $x=a$ from only one side (left or right). Left-hand limits approach from values of $x$ less than $a$, right-hand from values greater than $a$.
A two-sided limit exists if and only if both one-sided limits exist and are equal. If $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$, then $\lim_{x \to a} f(x) = L$. If they are not equal, the two-sided limit does not exist (DNE).
3. Infinite Limits and Limits at Infinity★★☆☆☆⏱ 3 min
Beyond limits at finite $x=a$, we estimate two common special limit types from graphs: infinite limits (the function approaches $\pm \infty$ as $x$ approaches a finite $a$, almost always at vertical asymptotes) and limits at infinity (the function approaches a $y$-value as $x$ approaches $\pm \infty$, describing end behavior).
For infinite limits at a vertical asymptote $x=a$: if both sides approach the same signed infinity, write $\lim_{x \to a} f(x) = +\infty$ or $-\infty$. If the two sides approach opposite infinities, the two-sided limit DNE. For limits at infinity: if the graph approaches a horizontal line $y=L$ as $x \to \pm \infty$, that $L$ is the limit, even if the function never actually reaches $L$.
4. AP-Style Concept Check★★★☆☆⏱ 4 min
Common Pitfalls
Why: Confuses the definition of a function value at a point with the definition of a limit, which describes behavior near the point, not at the point.
Why: Associates the minus sign with negative $y$ instead of approaching from values less than $a$, and plus with positive $y$ instead of values greater than $a$.
Why: Forgets the requirement that both one-sided limits must exist and be equal for a two-sided limit to exist.
Why: Mixes up input and output when reading limit questions.
Why: Believes the function must actually reach the limit value for the limit to exist.
Why: Assumes all non-finite limits are DNE, missing the context of signed infinite limits.