Why Compare Values at Different Distances?
When a table is used to estimate what a function does near an input \(a\), a single output is rarely enough. The useful evidence comes from comparing several input-output pairs: as the inputs get closer to \(a\), do their outputs appear to settle near one number?
The distances from the target matter. An input \(0.1\) unit from \(a\) is farther away than an input \(0.01\) unit from \(a\). Comparing their outputs, and then comparing still-closer inputs, helps distinguish a genuine pattern from a value that merely happens to be close once.
How to Read a Comparison Table
For each input \(x\), note its distance from the target \(a\), which is \(|x-a|\). Then compare outputs at successively smaller distances. If the table includes inputs on both sides of \(a\), keep the left-side and right-side values in separate groups at first. Each group can have its own pattern, so combining them too early can hide useful information.
Look at how the outputs change, but do not assume they must move steadily in one direction. The values may alternate above and below a candidate, or make small reversals, while still getting closer overall. A helpful check is to compare each output’s distance from a possible candidate value. That distance is the absolute difference between the output and the candidate.
Here \(L\) is a possible stable output suggested by the data. For example, if the outputs are \(4.8\), \(5.1\), and \(4.99\), they do not increase or decrease steadily. But all three are close to \(5\), and the last is especially close. The important question is not simply whether outputs change; it is whether they are becoming concentrated near one value as the inputs approach \(a\).
Worked Example: Matching Distances on Both Sides
Worked Example: Matching Distances on Both Sides
A table gives the following values near \(x=3\). Compare the outputs at matching distances and estimate a stable value.
| Input \(x\) | Distance \(|x-3|\) | Output \(f(x)\) |
|---|---|---|
| 2.9 | 0.1 | 2.1 |
| 3.1 | 0.1 | 2.9 |
| 2.99 | 0.01 | 2.46 |
| 3.01 | 0.01 | 2.54 |
| 2.999 | 0.001 | 2.496 |
| 3.001 | 0.001 | 2.504 |
Step 1: Compare the left-side outputs. As \(x\) moves from \(2.9\) to \(2.99\) to \(2.999\), the distances from \(3\) shrink from \(0.1\) to \(0.01\) to \(0.001\). The outputs are \(2.1\), \(2.46\), and \(2.496\). They move closer to \(2.5\).
Step 2: Compare the right-side outputs. At the same distances, the outputs are \(2.9\), \(2.54\), and \(2.504\). These also move closer to \(2.5\), now from above.
Step 3: Check the output distances from the candidate. For the left-side outputs, the distances from \(2.5\) are \(|2.1-2.5|=0.4\), \(|2.46-2.5|=0.04\), and \(|2.496-2.5|=0.004\). For the right-side outputs, they are \(|2.9-2.5|=0.4\), \(|2.54-2.5|=0.04\), and \(|2.504-2.5|=0.004\).
At each smaller input distance, the output is closer to \(2.5\). The table therefore gives strong numerical evidence that the outputs on each side are settling near \(2.5\). This conclusion describes the pattern in the displayed data; a finite table does not establish what happens at every nearby input.
Why Shrinking Output Differences Matter
A simple formula shows why comparing distances can reveal a stable value. Suppose outputs near \(x=a\) follow the exact rule \(f(x)=L+k(x-a)\), where \(L\) and \(k\) are constants. For an input \(d\) units to the right of \(a\), the output is \(L+kd\), so its distance from \(L\) is \(|k|d\). For an input \(d\) units to the left, the output is \(L-kd\), whose distance from \(L\) is also \(|k|d\).
Thus, when \(d\) is divided by \(10\), the output’s distance from \(L\) is also divided by \(10\). More generally, for any desired positive tolerance \(\varepsilon\), if \(k\ne 0\), choosing \(d<\varepsilon/|k|\) gives \(|k|d<\varepsilon\). If \(k=0\), the output equals \(L\) exactly for every such input. This is a complete argument for this particular linear rule: shrinking input distance forces shrinking output distance from \(L\).
A table can suggest a similar pattern, but its entries do not show that an unknown function follows this rule between the listed inputs. The calculation explains what stable behavior can look like; it does not turn a few table rows into a general proof.
Worked Example: Outputs Can Wobble and Still Get Close
Worked Example: Outputs Can Wobble and Still Get Close
The following data are recorded near \(x=1\). Decide whether the values appear to settle near a candidate number.
| Input \(x\) | Distance \(|x-1|\) | Output \(f(x)\) |
|---|---|---|
| 0.9 | 0.1 | 1.20 |
| 1.1 | 0.1 | 0.80 |
| 0.99 | 0.01 | 0.98 |
| 1.01 | 0.01 | 1.04 |
| 0.999 | 0.001 | 1.003 |
| 1.001 | 0.001 | 0.997 |
Step 1: Compare the left-side values. The outputs are \(1.20\), \(0.98\), and \(1.003\). They do not move steadily upward or steadily downward. Their distances from \(1\), however, are \(|1.20-1|=0.20\), \(|0.98-1|=0.02\), and \(|1.003-1|=0.003\), which decrease as the inputs get closer to \(1\).
Step 2: Compare the right-side values. The outputs are \(0.80\), \(1.04\), and \(0.997\). These also wobble around \(1\). Their distances from \(1\) are \(|0.80-1|=0.20\), \(|1.04-1|=0.04\), and \(|0.997-1|=0.003\).
Both sets of outputs become closer to \(1\) at the smaller distances shown, even though the values are not monotonic. The table supports \(1\) as a candidate approached value. The check is the output distance from \(1\), not whether every new output is larger or smaller than the preceding one.
When the Table Does Not Show a Stable Value
Sometimes outputs fail to cluster near any apparent finite number as the inputs get closer to the target. For instance, outputs of \(10\), \(100\), and \(1000\) at distances \(0.1\), \(0.01\), and \(0.001\) do not suggest that the outputs are settling near a fixed value. Instead, they grow as the input distances shrink.
This is a warning about the pattern in the table, not a proof about all possible inputs. A different set of unlisted inputs could behave differently. Numerical evidence should be reported with the scope it supports: the given entries do or do not show a trend toward a stable value.
Worked Example: A Target-Point Value Can Differ
Worked Example: A Target-Point Value Can Differ
Suppose a table near \(x=2\) includes the target itself as well as nearby inputs.
| Input \(x\) | Distance \(|x-2|\) | Output \(f(x)\) |
|---|---|---|
| 1.9 | 0.1 | 2.8 |
| 1.99 | 0.01 | 2.98 |
| 1.999 | 0.001 | 2.998 |
| 2 | 0 | 100 |
Step 1: Compare only the nearby inputs. The outputs at distances \(0.1\), \(0.01\), and \(0.001\) are \(2.8\), \(2.98\), and \(2.998\). Their distances from \(3\) are \(|2.8-3|=0.2\), \(|2.98-3|=0.02\), and \(|2.998-3|=0.002\).
Step 2: Interpret the target row carefully. The output at \(x=2\) is \(100\), but that row has distance zero and is not a nearby input different from the target. The three nearby rows get closer in output to \(3\), so the table supports \(3\) as the candidate approached value. The target-point output \(100\) does not change the pattern among those nearby rows.
Check: Each listed nearby output differs from \(3\) by the stated amount: \(3-2.8=0.2\), \(3-2.98=0.02\), and \(3-2.998=0.002\). The comparison is internally consistent.
A Reliable Comparison Routine
When reading a table, use a consistent sequence of checks:
Identify the input \(a\) whose nearby behavior is being examined.
Compare rows according to \(|x-a|\), from larger distances to smaller ones, rather than relying on the order in which the table happens to list them.
If there are inputs on both sides, compare each side's outputs separately before drawing an overall impression.
For a plausible value \(L\), compute or estimate \(|f(x)-L|\) for the listed outputs and check whether those differences tend to shrink.
Say that the table suggests or supports a value; do not claim that a finite number of rows proves the behavior for all nearby inputs.
Common Mistake: Assuming Outputs Must Move in One Direction
A common error is to reject a candidate just because the outputs reverse direction. For example, the sequence \(1.20,\ 0.98,\ 1.003\) first decreases and then increases. It would be incorrect to conclude from that reversal alone that the outputs are not settling near \(1\). The corresponding distances from \(1\) are \(0.20,\ 0.02,\ 0.003\), which steadily shrink.
The correction is to compare each output with the candidate, rather than compare outputs only with one another. Monotonic movement can be convenient, but it is not required for outputs to approach a stable value. Also avoid the opposite overclaim: even if all listed differences shrink, the data remain evidence from sampled inputs, not a guarantee about every input close to the target.
Check Your Understanding
Use distances from the target and output differences to explain what each table suggests. Remember to distinguish a table-based estimate from a proof.
- Near \(x=4\), outputs at distances \(0.1,\ 0.01,\ 0.001\) are \(6.9,\ 6.99,\ 6.999\). What candidate value do they support, and what are the outputs' distances from it?
- Near \(x=0\), one side has outputs \(2.2,\ 1.8,\ 2.03\) at distances \(0.1,\ 0.01,\ 0.001\). Do the outputs move monotonically? What candidate should you test?
- A table gives output \(9\) at the target and outputs \(4.7,\ 4.97,\ 4.997\) at distances \(0.1,\ 0.01,\ 0.001\). Which rows describe nearby behavior, and what value do they suggest?
- Why is a finite table not enough to prove what a function does at every input close to the target?
- What comparison can help when outputs wobble rather than move steadily up or down?
