Compare left and right estimates

Two tables, two answers

Table A lists outputs of one function near 0. The inputs are 0.1, 0.01 and 0.001 below 0 in the left column and above 0 in the right column.

Press Show the estimates. The orange number under each column is the number its outputs come close to, and in table A both estimates are 0.5.

Press Switch to table B. The left outputs come close to 1 and the right outputs come close to 3, so the estimates are different numbers.

When the two estimates are the same, one number describes both sides. When they are different, no single number does.

Any two columns

The table uses symbols in place of numbers. The left outputs come close to an estimate \(\cy{L}\), and at first the right outputs come close to \(\cy{L}\) too.

Press Make the right estimate different. The right outputs now come close to another number \(\cy{R}\), so the sign between the estimates changes from equal to unequal.

Add rows with the slider. Each output comes closer to its estimate, but the two estimates stay the same.

Compare the two estimates, and not the individual rows. One number fits both columns only when the estimates are equal.

Notation

1. The left column gives an estimate \(L\) of the limit from the left. The right column gives an estimate \(R\) of the limit from the right.

   \[\begin{aligned} \lim_{x \to c^-} f(x) &\approx \cy{L} \\ \lim_{x \to c^+} f(x) &\approx \cy{R} \end{aligned}\]

2. If \(L\) and \(R\) are the same number, the table suggests that the limit exists and equals that number.

   \[\cy{L} = \cy{R} \;\Longrightarrow\; \lim_{x \to c} f(x) \approx \cy{L}\]

3. If \(L\) and \(R\) are different numbers, the table suggests that the two-sided limit does not exist.

   \[\begin{aligned} \cy{L} \neq \cy{R} \;&\Longrightarrow\; \lim_{x \to c} f(x) \\ &\text{does not exist} \end{aligned}\]

4. To compare, both columns need inputs close to \(c\), at about the same distances. A table with one column gives nothing to compare.

   A table suggests the conclusion but does not prove it, because it shows only a few inputs.

When it fails

Suppose the left outputs come close to L and the right outputs come close to a different number R. Closer rows do not remove the gap between L and R, so no single number fits both sides.

If instead the gap between the two columns shrinks with each closer row, the estimates may still be equal. More rows are then needed to decide.

Example 1: opposite sides of the number

The table lists \(g(x) = \dfrac{e^x - 1}{x}\). Left column: (\(-0.1\), 0.9516), (\(-0.01\), 0.9950), (\(-0.001\), 0.9995). Right column: (0.1, 1.0517), (0.01, 1.0050), (0.001, 1.0005). Do the columns suggest that the limit at 0 exists?

1. In the left column, the outputs 0.9516, 0.9950 and 0.9995 rise and come close to 1.

   \[L \approx \cy{1}\]

2. In the right column, the outputs 1.0517, 1.0050 and 1.0005 fall and come close to 1.

   \[R \approx \cy{1}\]

3. The left outputs are below 1 and the right outputs are above 1, so the columns approach 1 from opposite sides. The two estimates are still the same number.

4. Because \(L = R = 1\), the table suggests that the limit exists and equals 1.

   \[\lim_{x \to 0} g(x) \approx \cy{1}\]

Example 2: a jump in the data

The table lists \(j(x)\). Left column: (1.9, \(-1.39\)), (1.99, \(-1.0399\)), (1.999, \(-1.0040\)). Right column: (2.1, 6.3), (2.01, 6.03), (2.001, 6.003). Do the columns suggest that the limit at 2 exists?

1. In the left column, the outputs \(-1.39\), \(-1.0399\) and \(-1.0040\) come close to \(-1\).

   \[L \approx \cy{-1}\]

2. In the right column, the outputs 6.3, 6.03 and 6.003 come close to 6.

   \[R \approx \cy{6}\]

3. A tempting wrong move: Each column comes close to a number, so the limit exists.

   The columns come close to different numbers, −1 and 6, but a limit is one number. The red bar shows the gap between the two heights, so no single number is close to both sides.

4. Because \(L \neq R\), the table suggests that the limit does not exist.

   \[\lim_{x \to 2} j(x) \text{ does not exist}\]

Practice

The table lists \(q(x) = \dfrac{2}{x}\). Left column: (\(-0.1\), \(-20\)), (\(-0.01\), \(-200\)), (\(-0.001\), \(-2000\)). Right column: (0.1, 20), (0.01, 200), (0.001, 2000). Do the columns suggest that the limit at 0 exists?

Press Add a closer row. Do the outputs in each column come close to a number?

Questions

Use the tables for the first two questions.

1. Table A lists outputs near \(x = 1\). What do the columns suggest?

   • The limit exists and is about 8.
   • The limit does not exist, because the left outputs are below 8 and the right outputs are above 8.
   • The limit is about 7.9.
   Answer

   The limit exists and is about 8.

2. Table B lists outputs near \(x = 1\). What do the columns suggest?

   • The limit does not exist.
   • The limit is about 2, halfway between 1 and 3.
   • The limit is about 1.
   Answer

   The limit does not exist.

3. Let \(r(x) = \dfrac{x^2 - 9}{|x - 3|}\). Compute \(r(2.99)\), \(r(2.999)\), \(r(3.001)\) and \(r(3.01)\). What do the outputs suggest about the limit at 3?

   • The limit does not exist.
   • The limit is about 6.
   • The limit is about 0.
   Answer

   The limit does not exist.

Summary

You read an estimate from each column and compare them: equal estimates suggest that the limit exists, and different ones suggest that it does not.

This works because a two-sided limit is one number that the outputs approach from both sides.

Use it when a table lists inputs on both sides of the target.

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