Wemazi

Topic 1.4

Read the trend from each side

Builds on recognizing unbounded behavior on a graph. 6 minutes.

A table in mixed order

The table lists the outputs of one function for inputs near 2. The rows are in mixed order, and there is no row for 2 itself, the input that the table approaches.

Press Sort by side. The inputs below 2 move to the left column, and the inputs above 2 move to the right. Each column runs from the farthest input to the nearest.

Press Show the trend and read down each column. On the left, the outputs 11.410, 11.940 and 11.994 rise toward 12. On the right, the outputs 12.610, 12.060 and 12.006 fall toward 12.

Each column gives its own estimate, which is the number that its outputs come close to. The orange arrows point to it, and here both estimates are 12.

Any table

The target input is now \(c\). The left column lists inputs below \(c\), and the right column lists inputs above \(c\).

Move the slider to add rows. Each new row has an input closer to \(c\), and the outputs in each column come closer to one number.

Press Show the estimates. The left outputs come close to \(\cy{L}\) and the right outputs to \(\cy{R}\). These can differ, so you estimate each side on its own. The table has no row for \(c\) itself.

Notation

  1. The left column lists inputs with \(x < c\), and the right column lists inputs with \(x > c\).

  2. Read down each column toward the input nearest \(c\). The outputs come close to a number, and that number is the estimate of the limit from that side.

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

    The sign \(\approx\) means "is close to". An estimate is not yet an exact value.

  3. The inputs must be close to \(c\), and they never equal \(c\). A limit asks about nearby inputs, not about \(c\) itself.

When it fails

If the outputs in a column grow larger with each closer row, as 10, 100 and 1000 do, they come close to no number. The red label shows that the column gives no estimate of a finite limit.

If the last rows still change by a large amount, add rows closer to the target before you estimate.

Example 1: two columns

The table lists \(h(x)\) for inputs near 4. Left column: (3.9, 0.2516), (3.99, 0.2502), (3.999, 0.2500). Right column: (4.1, 0.2485), (4.01, 0.2498), (4.001, 0.2500). Estimate the limit from each side at \(x = 4\).

  1. The left column lists inputs below 4. Read down it: 0.2516, 0.2502, 0.2500. The outputs fall and come close to 0.25.

    \[\lim_{x \to 4^-} h(x) \approx \cy{0.25}\]

  2. The right column lists inputs above 4. Read down it: 0.2485, 0.2498, 0.2500. The outputs rise and come close to 0.25.

    \[\lim_{x \to 4^+} h(x) \approx \cy{0.25}\]

  3. The orange rows, nearest to 4, decide each estimate. The farthest rows only show where each trend begins.

Example 2: a table in mixed order

The table lists \(w(x)\) in this order: (5.01, 4.005), (4.9, 1.9), (5.001, 4.0005), (4.99, 1.99), (5.1, 4.05), (4.999, 1.999). Estimate the limit from each side at \(x = 5\).

  1. Sort the rows by side. The left column holds the inputs 4.9, 4.99 and 4.999. The right column holds 5.1, 5.01 and 5.001.

  2. A tempting wrong move: The first output in the left column is 1.9, so the limit from the left is 1.9.

    \[\lim_{x \to 5^-} w(x) \approx 1.9\]

    The red row has the input farthest from 5. Down the column the outputs continue to rise, to 1.999 in the orange row nearest 5, so the trend goes past 1.9.

  3. Read down the whole left column. The outputs 1.9, 1.99 and 1.999 come close to 2.

    \[\lim_{x \to 5^-} w(x) \approx \cy{2}\]

  4. The right column gives 4.05, 4.005 and 4.0005. These outputs fall and come close to 4.

    \[\lim_{x \to 5^+} w(x) \approx \cy{4}\]

Practice

The table lists \(p(x)\) in this order: (3.01, \(-0.1107\)), (2.9, \(-0.1149\)), (3.001, \(-0.1111\)), (2.99, \(-0.1115\)), (3.1, \(-0.1075\)), (2.999, \(-0.1111\)). Estimate the limit from each side at \(x = 3\).

Sort the rows by side. Then read each column toward the input nearest 3.

Solution

  1. The left column holds 2.9, 2.99 and 2.999. Its outputs \(-0.1149\), \(-0.1115\) and \(-0.1111\) come close to \(-0.111\).

    \[\lim_{x \to 3^-} p(x) \approx \cy{-0.111}\]

  2. The right column holds 3.1, 3.01 and 3.001. Its outputs \(-0.1075\), \(-0.1107\) and \(-0.1111\) also come close to \(-0.111\).

    \[\lim_{x \to 3^+} p(x) \approx \cy{-0.111}\]

Questions

Use the table for the first two questions.

  1. What does the left column suggest for the limit from the left at \(x = 1\)?

    • About \(3\)
    • \(2.9\)
    • About \(5\)
    Answer

    About \(3\)

  2. What does the right column suggest for the limit from the right at \(x = 1\)?

    • About \(5\)
    • \(5.1\)
    • About \(3\)
    Answer

    About \(5\)

  3. Let \(t(x) = \dfrac{x^2 - 2x - 3}{x - 3}\). Compute \(t(2.9)\), \(t(2.99)\) and \(t(2.999)\). The outputs come close to which number?

    • \(4\)
    • \(3\)
    • \(0\)
    Answer

    \(4\)

Summary

You sort the rows by the side of the target input, and you read each column toward the input nearest the target.

This works because a one-sided limit depends only on the inputs on that side that come close to the target.

Use it when a table of values replaces a graph or a formula.

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