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Calculus BC · Unit 1 · 1.3
AP Calculus · Limits and Continuity
Estimating Limit Values from Graphs
Trace each branch and read where it is heading. The plotted point is a separate question.
key idea$\text{Read where the curve is HEADING, not the plotted point}$
Reading the graph
Trace the curve toward $x=c$ from the left, then from the right, and ask what height each branch is heading for. The filled dot at $x=c$ is $f(c)$ — it is a different question and you should ignore it while reading the limit.
limit exists, value differs
| read off the graph | left panel | middle panel | right panel |
|---|
| $\lim_{x\to2^{-}}f$ | $3$ | $1$ | $4$ |
| $\lim_{x\to2^{+}}f$ | $3$ | $3.4$ | $4$ |
| $\lim_{x\to2}f$ | $3$ | DNE | $4$ |
| $f(2)$ | $1$ | $3.4$ | undefined |
Four separate readings per graph. The bottom row never affects the row above it.
reading order
- Trace each branch separately before combining.
- An open circle marks the limit; a filled dot marks the value.
- Different heights on the two branches ⇒ the two-sided limit is DNE.
- Match your decimals to the gridlines you can actually resolve.
| Symbol on the graph | What it tells you | Does it affect the limit? |
|---|
| open circle at $(c,L)$ | the curve heads for $L$ but $f(c)\neq L$ | no — the limit is still $L$ |
| filled dot at $(c,k)$ | $f(c)=k$ | no — that is the value, not the limit |
| branches at different heights | the one-sided limits disagree | yes — the limit is DNE |
| branches shooting off the top | unbounded growth | yes — DNE, describe as $\pm\infty$ |
| nothing drawn at $x=c$ | $f(c)$ is undefined | no — the limit can still exist |
Only the middle two rows can destroy a limit. Everything to do with the plotted point is a separate question.
Worked examples
EX 1
Limit and value disagree
left panelFrom the left panel: find $\lim\limits_{x\to2}f(x)$ and $f(2)$.
1
Trace from the left
The branch climbs toward height $3$ as $x$ approaches $2$.
2
Trace from the right
The branch continues from height $3$ upward. Both sides head for $3$.
3
State the limit
The one-sided limits agree, so $\lim_{x\to2}f(x)=3$. The open circle at $(2,3)$ marks where the curve is heading.
4
Read the value separately
The filled dot sits at $(2,1)$, so $f(2)=1$.
5
Answer both parts
$\lim_{x\to2}f(x)=3$ and $f(2)=1$. They differ — that is exactly why the two questions are asked separately.
EX 2
A jump
middle panelFrom the middle panel: evaluate the one-sided limits at $x=2$ and the two-sided limit.
1
Left branch
For $x$ just below $2$ the curve sits at height $1$, so $\lim_{x\to2^{-}}f(x)=1$.
2
Right branch
For $x$ just above $2$ it sits at about $3.4$, so $\lim_{x\to2^{+}}f(x)=3.4$.
3
Compare
$1\neq3.4$ — the branches head for different heights.
4
Conclude
$\lim_{x\to2}f(x)$ does not exist. Note that both one-sided limits do exist; it is only the two-sided limit that fails.
EX 3
No value at all
right panelFrom the right panel: find $\lim\limits_{x\to2}f(x)$ and $f(2)$.
1
Trace both sides
The parabola rises to a peak; both branches head for height $4$.
2
State the limit
$\lim_{x\to2}f(x)=4$.
3
Look for a plotted point
There is an open circle and no filled dot, so $f(2)$ is undefined.
4
Do not let that change the limit
A missing value cannot stop the curve from heading somewhere. The limit exists and equals $4$; the function is simply not defined there.
EX 4
How precise can you be?
estimationA graph is drawn on a grid with unlabelled minor gridlines every $0.2$. Reading a limit, a student writes $\lim_{x\to1}f(x)=2.37$. Comment.
1
Ask what the grid supports
With gridlines every $0.2$, you can reliably read to about half a gridline — roughly $\pm0.1$.
2
Judge the claim
Two decimal places implies precision of $\pm0.005$, which the graph cannot support. $2.37$ is false precision.
3
Give a defensible answer
“approximately $2.4$”, or a bracket such as “between $2.2$ and $2.6$”.
4
Note the wording
The CED says estimate for graphs and tables. An exact-looking answer from a picture claims more than the evidence gives.
Watch out
The filled dot is not the limit
$f(c)$ and $\lim_{x\to c}f(x)$ are different questions with different answers. Read the limit by tracing the curve, then look at the dot separately.
Both one-sided limits can exist while the limit does not
A jump has two perfectly good one-sided limits. “DNE” refers to the two-sided limit only — say which one you mean.
An open circle is information, not a gap
It marks precisely the height the curve is heading for. It confirms the limit rather than threatening it.
Do not over-read the grid
Graph readings are estimates. Match your decimal places to the gridlines you can actually resolve.