Calculus BC · Unit 1: Limits and Continuity · 14 min read · Updated 2026-05-10

Estimating limit values from graphs — AP Calculus BC

AP Calculus BC · Unit 1: Limits and Continuity · 14 min read

1. One-Sided Limits from Graphs ★☆☆☆☆ ⏱ 3 min

To estimate a one-sided limit from a graph, trace the graph from the direction of the approach, and read the $y$-value the graph approaches as it nears $x=a$. Open circles, closed circles, and undefined points do not change the limit—only the trend of the graph near $a$ from the given side matters.

📐 Worked Example

The graph of $f(x)$ has a jump discontinuity at $x=4$: for $x<4$, the graph approaches an open circle at $(4, 1)$; for $x>4$, the graph approaches an open circle at $(4, -5)$; and $f(4)=1$ is marked by a closed circle at $(4,1)$. Estimate $\lim_{x \to 4^-} f(x)$ and $\lim_{x \to 4^+} f(x)$.

Approach $x=4$ from the direction of $x<4$, tracing the graph toward $x=4$.
The graph approaches a $y$-value of $1$ at the open circle $(4,1)$, so the left-hand limit equals $1$.
The closed circle at $(4,1)$ matches the approached value, but the result would be identical even if the closed circle were placed at a different $y$-value.
For the right-hand limit, approach $x=4$ from the direction of $x>4$, tracing the graph toward $x=4$.
The graph approaches a $y$-value of $-5$ at the open circle $(4,-5)$, so the right-hand limit equals $-5$.
Final estimates:
$\lim_{x \to 4^-} f(x) = 1 \quad \text{and} \quad \lim_{x \to 4^+} f(x) = -5$

Exam tip: On AP FRQs, always explicitly mention 'approaching from the left/right' in your justification for a one-sided limit to earn full points.

2. Two-Sided Limits from Graphs ★★☆☆☆ ⏱ 4 min

\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L

To estimate a two-sided limit from a graph, always first estimate both one-sided limits, then check for agreement. If they match, that matching value is your two-sided limit. If they do not match, the two-sided limit does not exist (DNE). Limits can exist at points where $f(a)$ is undefined (a hole/point discontinuity) or where $f(a)$ is defined at a different $y$-value than the limit.

Exam tip: If an MCQ option says a limit does not exist, double-check that the one-sided limits actually do not match. Many students pick DNE incorrectly when limits match but the function is undefined at the point.

3. Infinite Limits and Limits at Infinity ★★☆☆☆ ⏱ 4 min

Two additional common limit types estimated from graphs are infinite limits (near vertical asymptotes) and limits at infinity (end behavior).

Exam tip: If an AP question asks 'does the limit exist' for an infinite limit, you must answer no. Writing $\lim_{x \to a} f(x) = \infty$ describes behavior, but it does not mean the limit exists as a finite value.

4. AP-Style Practice Check ★★☆☆☆ ⏱ 3 min

📐 Worked Example

The graph of $g(x)$ has the following key features: Vertical asymptote at $x=1$; Hole at $x=3$ at $(3, 5)$, with $g(3) = 1$ marked by a closed circle at $(3,1)$; Horizontal asymptote at $y=2$ as $x \to \infty$. (a) Estimate $\lim_{x \to 3} g(x)$. Justify your answer. (b) As $x \to 1^-$, $g(x)$ increases without bound, and as $x \to 1^+$, $g(x)$ decreases without bound. Write the one-sided limits using correct notation, and state whether $\lim_{x \to 1} g(x)$ exists as a finite number. (c) Estimate $\lim_{x \to \infty} g(x)$ and explain what this means in terms of the graph's end behavior.

(a) As $x$ approaches 3 from both the left and right, the graph approaches the hole at $(3,5)$, so both one-sided limits equal 5. Since the one-sided limits are equal:
$\lim_{x \to 3} g(x) = 5$
The value $g(3)=1$ does not change the limit, because limits depend on behavior near $x=3$, not at $x=3$.
(b) The one-sided limits are written as:
$\lim_{x \to 1^-} g(x) = \infty \quad \text{and} \quad \lim_{x \to 1^+} g(x) = -\infty$
The one-sided limits do not agree and neither is finite, so $\lim_{x \to 1} g(x)$ does not exist as a finite number.
(c) As $x$ grows without bound to the right, the graph approaches the horizontal asymptote at $y=2$, so:
$\lim_{x \to \infty} g(x) = 2$
This means that as $x$ gets larger and larger, the value of $g(x)$ gets arbitrarily close to 2.

Common Pitfalls

Why: Students confuse the actual value of the function at $a$ with the value the function approaches near $a$.

Why: Students incorrectly assume a limit can't exist if the function doesn't exist at the point.

Why: Students confuse the position of the closed circle (the function value) with the limit of the graph's trend.

Why: Students think labeling the behavior as infinity means the limit exists.

Why: Students use the nearest visible point instead of following the end behavior.

Quick Reference Cheatsheet

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