Separate a reading from a limit
A table with a row for the target
A reading is the output of a function at one input. The upper part of the table lists readings at inputs near the target 2, and the lower row lists the reading at 2 itself.
Read down each column of the upper part. The outputs come close to \(\cy{-1}\), the orange estimate under each column, but the reading at 2 is 4.
Move the slider for the reading at 2. The nearby rows do not change, so the number that they approach stays \(-1\).
Press Remove the row for 2, and the nearby rows still approach \(-1\). Press Add a closer row, and the new outputs \(-1.0001\) and \(-0.9999\) are closer still.
The limit is the number that the readings approach as the inputs come close to the target. The reading at the target is not one of those readings.
Any table
The upper rows approach a number \(\cy{L}\), and the lower row shows the reading at the target \(c\).
Move the lower reading. The number \(L\) stays where it is, because the nearby rows decide it.
Add a closer row. The new reading is closer to \(L\), but it is still not \(L\).
The limit is the number that the nearby readings approach. It is not the reading at the target, and it is not any single nearby reading.
Notation
The reading at the target \(c\) is written \(f(c)\).
\[f(c)\]If the table has no row for \(c\), the table does not give \(f(c)\).
The limit is the number that the readings at nearby inputs approach, so it does not use the row for \(c\).
\[\lim_{x \to c} f(x) \approx \cy{L}\]A nearby reading is close to the limit, but it is not the limit. The nearby rows must come close to one number for the limit to exist.
The reading \(f(c)\) may equal the limit, it may differ from it, or \(f\) may be undefined at \(c\).
When it fails
Move the reading at \(c\). The reading \(f(c)\) changes, but the limit stays \(L\), because the nearby rows decide it.
Copying \(f(c)\) as the limit is correct only when the two numbers are equal. You know this only after you read the nearby rows.
Example 1: a reading that differs
The table lists \(j\) near 4. Left column: (3.9, 10.7), (3.99, 10.97), (3.999, 10.997). Right column: (4.1, 11.3), (4.01, 11.03), (4.001, 11.003). The reading at 4 is \(j(4) = 0\). Find the reading at 4 and estimate the limit at 4.
The reading at the target 4 is in the lower row.
\[j(4) = \cy{0}\]The rows near 4 are separate readings. The left outputs 10.7, 10.97 and 10.997 rise toward 11. The right outputs 11.3, 11.03 and 11.003 fall toward 11.
Both columns come close to 11, so the estimate of the limit is 11.
\[\lim_{x \to 4} j(x) \approx \cy{11}\]The reading 0 and the limit 11 are different, because the limit uses only the rows near 4.
Example 2: no row for the target
The table lists \(v\) near 0, and it has no row for 0. Left column: (\(-0.1\), 1.1111), (\(-0.01\), 1.0101), (\(-0.001\), 1.0010). Right column: (0.1, 0.9091), (0.01, 0.9901), (0.001, 0.9990). Estimate the limit at 0.
The nearest rows are \(x = -0.001\) and \(x = 0.001\), with outputs 1.0010 and 0.9990.
A tempting wrong move: The nearest reading on the right is 0.9990, so the limit is 0.9990.
\[\lim_{x \to 0} v(x) \approx 0.9990\]The red row is one reading, and the closer input 0.0001 gives 0.9999, which is nearer to 1. The left outputs stay above 1 and the right outputs stay below 1, so no single reading is the number that both columns approach.
Read the trend in each column. The left outputs fall toward 1, and the right outputs rise toward 1.
\[\lim_{x \to 0} v(x) \approx \cy{1}\]No row in the table has the output 1, so this limit is not equal to any reading.
Practice
The table lists \(z\) near 6, and the reading is \(z(6) = 9\). Left column: (5.9, 8.8), (5.99, 8.98), (5.999, 8.998). Right column: (6.1, 9.2), (6.01, 9.02), (6.001, 9.002). State the reading at 6 and estimate the limit at 6. Are the two numbers equal?
Read the row for 6 first. Then read the rows near 6 as two columns.
Solution
The reading at 6 is the lower row, \(z(6) = 9\).
\[z(6) = \cy{9}\]The left outputs 8.8, 8.98 and 8.998 rise toward 9. The right outputs 9.2, 9.02 and 9.002 fall toward 9.
\[\lim_{x \to 6} z(x) \approx \cy{9}\]The two numbers are equal here, but you know this only because you read the nearby rows, not the row for 6.
Questions
Use the table for the first two questions.
What is the reading at \(x = 3\)?
- 2
- 5
- 4.996
Answer
2
The readings near \(x = 3\) come close to which number?
- 5
- 2
- 4.996
Answer
5
Let \(g(x) = x^3 - 2\) for \(x \neq 2\), and let \(g(2) = 0\). Compute \(g(1.99)\) and \(g(2.01)\). They are close to which number?
- 6
- 0
- 5.88
Answer
6
Summary
You read the output at the target as one reading, and the limit as the number that the readings at nearby inputs approach.
This works because a limit uses only inputs close to the target, never the target itself.
Use it when a table lists the reading at the target next to readings near it.
This is one of four skills in topic 1.4 of Which & Why: AP Calculus.