How can a table of values suggest a limit without proving it?
Quick answer
Choose inputs just below and just above the target, calculate the function values, and look for a shared trend. For example, values of (x²−1)/(x−1) near x = 1 approach 2 from both sides. That suggests a limit of 2, but a finite table checks only selected inputs; it cannot prove what happens for every sufficiently close input.
The question, the details, the next step.
I have a function and want to estimate its limit from a table. What should I put in the table?
Write down the input you are approaching—in this example, x = 1. Choose several inputs slightly below 1 and several slightly above 1, then calculate the function value at each input. Keep the two sides separate so you can compare their trends.
Consider f(x) = (x² − 1)/(x − 1), for x ≠ 1. Since x² − 1 = (x − 1)(x + 1), this simplifies to f(x) = x + 1 whenever x ≠ 1. Here is a table:
| Side of 1 | x | f(x) |
|---|---|---|
| Below | 0.9 | 1.9 |
| Below | 0.99 | 1.99 |
| Above | 1.01 | 2.01 |
| Above | 1.1 | 2.1 |
The inputs closer to 1 give function values closer to 2, from below on the left and above on the right. So the table suggests that the limit is 2. Choosing inputs approaching the target from both sides and examining the resulting values is a useful way to estimate a limit [S1].
Why do I need values on both sides? If the values below 1 are heading toward 2, isn't that enough?
Values below 1 show only what the function appears to do as x approaches 1 from the left. Values above 1 check the approach from the right. For a two-sided limit, you want both sides to point toward the same value. If the sides appear to approach different values, the table does not support a shared two-sided limit.
In this example, the left-side values are below 2 and the right-side values are above 2; both trends point toward 2. Also, the original expression is undefined at x = 1 because its denominator is zero. That does not prevent us from examining nearby inputs: a limit concerns what happens as x gets close to the target, not necessarily the function’s value at the target.
The numbers are close to 2. Does that prove the limit is 2?
No. A table contains only finitely many samples. Even many values close to 1 cannot, by themselves, establish how the function behaves at every sufficiently close input. Tables can provide insight and help estimate a limit, but they are not the formal definition or proof [S1].
For this example, algebra gives a separate justification: for x ≠ 1, the expression simplifies to x + 1, which approaches 2 as x approaches 1. The table suggests the result; the algebra supports it. If a task asks for proof, use an appropriate mathematical argument rather than presenting the table alone as proof.
How can I write up the conclusion without overstating what the table shows?
State the target, describe what the values from each side appear to do, and call the result a suggestion or estimate unless you have a separate proof. For this example, you could write:
“Using inputs below and above 1, the function values appear to approach 2 from both sides. The table suggests that the limit as x approaches 1 is 2. This is numerical evidence, not a proof; for x ≠ 1, the expression simplifies to x + 1, which provides an algebraic justification for this example.”
That wording separates the observation, the conjecture, and the justification. If the two sides do not appear to agree, report that rather than choosing a value based on only one side.
A practical checklist
Write the target input clearly before choosing sample values.
Choose inputs on both sides of the target, with some closer than others.
Calculate the function values and label which side each input comes from.
Check whether both sides appear to approach the same number.
Describe a table-based result as a suggestion or estimate, not a proof.
Use a separate mathematical argument when a proof is required.
A few more useful answers
Should I include the target input itself in the table?
You can note whether the function is defined there, but the limit concerns values at nearby inputs. In this example, the original expression is undefined at x = 1, yet the table still suggests a limit.
What if the left and right sides seem to approach different values?
Then the table does not support a shared two-sided limit. Keep the sides separate and investigate further rather than reporting one side’s value as the two-sided result.
Can adding more decimal places make the table a proof?
No. More precise calculations may make a numerical pattern easier to inspect, but a table still checks only selected inputs. It does not establish behavior for every sufficiently close input.
Sources
References used for this guide.
- [S1] OpenStax Calculus Volume 1: The Limit of a Function openstax.org · Checked