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Calculus AB · Unit 1 · 1.4
AP Calculus · Limits and Continuity
Estimating Limit Values from Tables
Close in from both sides. Agreement is evidence — and evidence is not proof.
key idea$\text{Approach }c\text{ from BOTH sides. Agreement is the evidence.}$
The method
A table gives you a sequence of ever-closer inputs. Read down each side and ask what number the outputs are settling on. Evidence from one side alone is not enough — a jump looks perfectly convincing until you check the other side.
| $x$ | $1.9$ | $1.99$ | $1.999$ | $\to 2 \leftarrow$ | $2.001$ | $2.01$ | $2.1$ |
|---|
| $f(x)$ | $3.9$ | $3.99$ | $3.999$ | ? | $4.001$ | $4.01$ | $4.1$ |
Both sides close in on $4$, so the evidence supports $\lim_{x\to2}f(x)\approx4$. The centre column is deliberately blank — a limit never uses the value at $c$.
| What the table shows | Reading | Answer |
|---|
| both sides settle on the same number | the limit exists | that number (as an estimate) |
| the two sides settle on different numbers | a jump | DNE |
| values grow without bound | unbounded | DNE; describe as $\pm\infty$ |
| values keep swinging with no pattern | oscillation | DNE |
| only one side is tabulated | insufficient evidence | a one-sided limit only |
A table can never prove a limit — it only supports an estimate. Say “approximately” unless you have algebra to confirm it.
Worked examples
EX 1
Both sides agree
$\dfrac{x^{2}-4}{x-2}$Use the table above, generated by $f(x)=\dfrac{x^{2}-4}{x-2}$, to estimate $\lim\limits_{x\to2}f(x)$. Then confirm algebraically.
1
Read the left side
$3.9,\;3.99,\;3.999$ — climbing toward $4$.
2
Read the right side
$4.1,\;4.01,\;4.001$ — descending toward $4$.
3
Compare
Both sides close in on the same number, so the estimate is $\lim_{x\to2}f(x)\approx4$.
4
Confirm with algebra
$\dfrac{(x-2)(x+2)}{x-2}=x+2$ for $x\neq2$, so the limit is exactly $4$.
5
Note the upgrade
The table gave an estimate; the algebra gave certainty. That is the right division of labour between the two.
EX 2
One side is not enough
a jumpA student tabulates $g$ only for $x=2.1,\,2.01,\,2.001$, gets $5.1,\,5.01,\,5.001$, and concludes $\lim_{x\to2}g(x)=5$. What is missing?
1
Name what was shown
The evidence supports $\lim_{x\to2^{+}}g(x)=5$ — the right-hand limit only.
2
State what is unknown
Nothing has been tested for $x\lt 2$. The left branch could settle anywhere.
3
Give a counterexample
If $g(x)=1$ for all $x\lt 2$, the left limit is $1$, the two sides disagree, and $\lim_{x\to2}g(x)$ does not exist — consistent with every value tabulated.
4
Fix it
Add $x=1.9,\,1.99,\,1.999$. Only if those also close in on $5$ does the two-sided estimate hold.
EX 3
A table that misleads
oscillationFor $h(x)=\sin\!\left(\tfrac{\pi}{x}\right)$, a student tabulates $x=1,\,\tfrac12,\,\tfrac13,\,\tfrac14$ and finds $h=0$ every time. They conclude $\lim_{x\to0}h(x)=0$. Evaluate.
1
Check the arithmetic
At $x=\tfrac1n$, $\tfrac{\pi}{x}=n\pi$ and $\sin(n\pi)=0$. Every tabulated value is genuinely $0$.
2
Spot the sampling problem
Those inputs land exactly on the zeros of the sine. The table samples one family of points, not the general behaviour.
3
Sample differently
At $x=\tfrac{2}{5}$, $\tfrac{\pi}{x}=\tfrac{5\pi}{2}$ and $h=1$. At $x=\tfrac{2}{7}$, $h=-1$.
4
Conclude
$h$ oscillates between $-1$ and $1$ infinitely often near $0$, so $\lim_{x\to0}h(x)$ does not exist.
5
The warning
A table shows only the points you chose. Agreement across a handful of inputs is evidence, never proof.
EX 4
Reading a rate from a table
appliedA tank's volume $V$ (litres) is recorded at times $t$ near $t=3$ min: $V(2.9)=48.2$, $V(2.99)=48.92$, $V(3.01)=49.08$, $V(3.1)=49.8$. Estimate $\lim\limits_{t\to3}V(t)$ and say what it means.
1
Left side
$48.2$, then $48.92$ — rising toward roughly $49$.
2
Right side
$49.8$, then $49.08$ — falling toward roughly $49$.
3
Estimate
Both sides close in on about $49$, so $\lim_{t\to3}V(t)\approx49$ litres.
4
Interpret with units
The volume approaches about $49$ litres as the time approaches $3$ minutes.
5
Judge the precision
The closest readings differ in the second decimal, so quoting $49.0$ is defensible; quoting $49.00$ is not.
How wrong can it be?
A table supports an estimate, never a proof
Finitely many points cannot rule out what happens between them. Use the word approximately unless algebra confirms the value.
Always tabulate both sides
One-sided data can only give a one-sided limit. A jump is invisible from one direction — Example 2.
Beware inputs that land on a pattern
Sampling $x=\tfrac1n$ for $\sin\!\left(\tfrac{\pi}{x}\right)$ hits every zero and hides the oscillation entirely.
Match precision to the data
If the closest tabulated values agree to two decimals, do not quote four. False precision is a real deduction.