Read what a limit statement means

Outputs near a missing point

The curve shows \(f(x) = \dfrac{\sqrt{x+4} - 2}{x}\). It has an open ring at \(x = 0\), because the formula gives 0 divided by 0 there, so \(f(0)\) has no value.

Two orange dots lie on the curve, one on each side of the ring. The height of each dot is its output, marked by an orange tick on the vertical axis.

Move the right dot toward \(x = 0\), and its output becomes 0.2485 at \(x = 0.1\) and 0.2498 at \(x = 0.01\). On the left, the output is 0.2516 at \(x = -0.1\) and 0.2502 at \(x = -0.01\).

From both sides the outputs approach 0.25, which is called the limit of \(f(x)\) as \(x\) approaches 0. The limit describes where the outputs go, so it does not need a value at \(x = 0\).

As close as you want

The input \(x\) approaches a number \(c\), and the output \(f(x)\) approaches a number \(L\). The open ring at \((c, L)\) shows that \(f\) may have no value at \(c\).

The orange band shows how close to \(L\) you want the outputs. The faint strip shows the inputs near \(c\). Where the curve is inside the strip but outside the band, it is red, because those outputs are too far from \(L\).

Make the band narrower, and red appears, because the strip is now too wide. Then make the strip narrower until the red disappears.

For every band, some strip is narrow enough that no red is left, so every output in it, except at \(c\), lies inside the band. This is what it means for \(f(x)\) to approach \(L\).

Notation

1. The statement says that as \(x\) approaches \(c\), the output \(f(x)\) approaches \(L\). The two orange dots move toward \(c\) from both sides.

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

2. The number after the arrow, \(\cq{c}\), is the input that \(x\) approaches, and the faint vertical line marks it.

3. The number after the equal sign, \(\cy{L}\), is the output that \(f(x)\) approaches, and the orange horizontal line marks it.

4. The input \(x\) approaches \(c\) but does not equal \(c\), so the statement says nothing about \(f(c)\). The statement needs \(f\) to have values on both sides of \(c\), close to \(c\).

When it fails

The outputs must approach one number from both sides. If the right part of the curve moves up, the left outputs still approach L, but the right outputs approach a larger number.

Inside the faint strip, the right part of the curve now lies outside the orange band, so it is red. The two sides approach different numbers, so the limit does not exist.

Example 1: reading a statement

Read the statement \(\lim_{x \to -2}(x^2 + 1) = 5\). Name the number that \(x\) approaches and the number that \(f(x)\) approaches. Then test the statement with inputs on both sides of \(-2\).

1. Here \(f(x) = x^2 + 1\), and the number after the arrow is \(-2\), so \(x\) approaches \(-2\). The number after the equal sign is 5, so \(f(x)\) approaches 5.

2. From the left, \(x = -2.1\) gives 5.41 and \(x = -2.01\) gives 5.0401. From the right, \(x = -1.9\) gives 4.61 and \(x = -1.99\) gives 4.9601.

3. The outputs from both sides approach 5, so the statement is correct.

   \[\lim_{x \to -2}(x^2 + 1) = \cy{5}\]

Example 2: finding a limit from a table

Find \(\lim_{x \to 2} x^3\). A table of \(f(x) = x^3\) gives \(f(1.9) = 6.859\), \(f(1.99) \approx 7.8806\), \(f(1.999) \approx 7.988\), \(f(2.1) = 9.261\), \(f(2.01) \approx 8.1206\) and \(f(2.001) \approx 8.012\).

1. The input \(x\) approaches 2. Each column lists the inputs on one side of 2, from the farthest to the nearest.

2. A tempting wrong move: The last output on the left is 7.988, so the limit is 7.988.

   \[\lim_{x \to 2} x^3 = 7.988\]

   The outputs on the left continue to rise as x moves closer to 2, and at x = 1.9999 the output is about 7.9988. The outputs pass the row outlined in red, so 7.988 is not the limit.

3. The outputs from the left rise toward 8, and the outputs from the right fall toward 8, so the limit is 8.

   \[\lim_{x \to 2} x^3 = \cy{8}\]

Practice

Estimate \(\lim_{x \to 0} \dfrac{\sqrt{x+1} - 1}{x}\). Use \(x = \pm 0.1\) and \(x = \pm 0.01\). Say which number \(x\) approaches and which number \(f(x)\) approaches.

Move both dots toward the ring. Then compare the heights of the two orange ticks on the vertical axis.

Questions

The graph shows \(g(x) = 4 - (x - 2)^2\) for \(x\) from 0 to 4.

1. As \(x\) approaches 2, which number does \(g(x)\) approach?

   • 4
   • 2
   • 0
   Answer

   4

2. Which statement is true about this graph?

   • \(\lim_{x \to 3} g(x) = 3\)
   • \(\lim_{x \to 3} g(x) = 4\)
   • \(\lim_{x \to 1} g(x) = 1\)
   Answer

   \(\lim_{x \to 3} g(x) = 3\)

3. Estimate \(\lim_{x \to 5} \dfrac{x^2 - 25}{x - 5}\) from the outputs at \(x = 4.99\) and \(x = 5.01\).

   • 10
   • 9.99
   • 5
   Answer

   10

Summary

The number after the arrow is the input that x approaches, and the number after the equal sign is the output that f(x) approaches.

You read the limit from the outputs on both sides, because the limit does not depend on the output at the input itself.

A limit statement describes what a function does near an input, even where the function has no value.

This is one of four skills in topic 1.2 of Which & Why: AP Calculus.