State what the data supports

Five readings, all zero

The table lists five readings of one function at inputs that come closer to 0. Every output is 0, so the table suggests that the limit as \(x\) approaches 0 is 0.

Press Add two readings. The new inputs, \(\tfrac{2}{5}\) and \(\tfrac{2}{9}\), each lie between two inputs already in the table, and both outputs are 1.

The function is \(f(x) = \sin\dfrac{\pi}{x}\). It swings between \(-1\) and 1 more and more often as \(x\) approaches 0, and the first five inputs are places where it crosses zero.

A table shows what the function does at the listed inputs. It does not show what the function does between them.

Any table

The orange dots are readings of a function at inputs near \(\cq{c}\), and their heights come close to a number \(\cy{L}\).

Press Draw another curve through the same readings. The faint dashed curve passes through every dot, but between the dots it swings far above and below the first curve.

Press Add a closer reading. The new dot also lies on both curves, so it cannot tell them apart.

A finite set of readings fits many different curves, so the data support an estimate of the limit but cannot prove an exact value.

Notation

1. An estimate says that the outputs come close to \(L\). The data support this statement, so you write it with the sign \(\approx\).

   \[\lim_{x \to c} f(x) \approx \cy{L}\]

2. An exact claim says that the limit equals \(L\), which is a statement about every input near \(c\) and not only the listed inputs.

   \[\lim_{x \to c} f(x) = \cy{L}\]

3. A table, a list of readings or a graphing window supports an estimate. An exact claim needs the formula and a method, such as algebra or a known limit.

4. Write the conclusion with the words that the data allow, such as suggests or is about. Use equals only for a claim that a method proves.

When it fails

A calculator keeps a fixed number of digits, so for a tiny input it can round the output to a wrong value, such as 0.

A row that breaks the trend of the earlier rows, like the red row, is a warning. Check that each row continues the trend before you use it.

Example 1: an estimate, not an exact value

The table lists \(k\) near 2. Left column: (1.9, 0.1671), (1.99, 0.1667), (1.999, 0.1667). Right column: (2.1, 0.1662), (2.01, 0.1666), (2.001, 0.1667). Which statement does the table support: A, the limit at 2 is about 0.167, or B, the limit at 2 equals 0.1667?

1. Both columns come close to 0.167, so the table supports statement A.

   \[\lim_{x \to 2} k(x) \approx \cy{0.167}\]

2. Statement B claims an exact value, but six readings to four decimals cannot prove one.

3. The formula and algebra from a later topic give the exact value \(\tfrac{1}{6}\). This is not 0.1667, so statement B is also false.

   As a check, \(\tfrac{1}{6} = 0.1666\ldots\), which matches the estimate.

Example 2: a row that breaks the pattern

A calculator that keeps 10 digits gives these readings of \(w(x) = \dfrac{\sqrt{x^2 + 16} - 4}{x^2}\). The left rows match the right rows. (0.1, 0.124980), (0.01, 0.125000), (0.001, 0.125000), (0.00001, 0.000000). Estimate the limit at 0.

1. In the first three rows of each column, the outputs come close to 0.125.

2. A tempting wrong move: The row nearest 0 is the last row, and it shows 0.000000. The limit is 0.

   \[\lim_{x \to 0} w(x) \approx 0\]

   The red rows drop to 0 after a steady trend toward 0.125, which is a warning. At \(x = 10^{-5}\), \(x^2 + 16 = 16.0000000001\), and a 10-digit calculator rounds this to 16. The numerator becomes \(\sqrt{16} - 4 = 0\).

3. The last row is a rounding error, so you estimate the limit from the rows that continue the trend.

   \[\lim_{x \to 0} w(x) \approx \cy{0.125}\]

4. Algebra, not a calculator, gives the exact limit \(\tfrac{1}{8} = 0.125\), which matches the steady rows.

Practice

A table lists \(p\) at the inputs 2.9, 2.99, 2.999 and 3.1. The outputs are 5.41, 5.9401, 5.994001 and 6.61. What does the table support about the limit at 3?

Sort the inputs by side. Count how many rows are on each side of 3.

Questions

Use the table for the first two questions.

1. Which statement does the table support?

   • The limit at 1 is about 2.
   • The limit at 1 equals 2, and the table proves it.
   • The limit at 1 is 1.9995.
   Answer

   The limit at 1 is about 2.

2. A student says that the table proves that the limit at 1 equals 2. What is wrong with this claim?

   • The table lists only six inputs, so it supports an estimate and cannot prove an exact value.
   • The right column is farther from 1 than the left column.
   • The outputs 1.9487 and 2.0488 are different, so the table suggests no limit.
   Answer

   The table lists only six inputs, so it supports an estimate and cannot prove an exact value.

3. Let \(p(x) = \dfrac{\sqrt{x + 4} - 2}{x}\). Compute \(p(0.1)\) and \(p(0.01)\). What do these two readings support?

   • The limit from the right is about 0.25. The two readings do not prove an exact value.
   • The limit at 0 equals 0.2498.
   • The limit does not exist, because the two readings are different.
   Answer

   The limit from the right is about 0.25. The two readings do not prove an exact value.

Summary

You describe what a table shows with the words about and suggests, and you use equals only for a claim that a method proves.

This works because a table shows the function only at the listed inputs, and not between them.

Use it when a conclusion comes from a table or a calculator.

This is one of four skills in topic 1.4 of Which & Why: AP Calculus.