What a limit is: graphs and tables

When you reach for it

To know what a function does at an input, you evaluate it. Often the question is what it does near the input. The formula \displaystyle \frac{x^2 - x - 6}{x - 3} refuses x = 3, yet its outputs near 3 crowd around one height. In Rates at an instant from shrinking intervals a speed at an instant was estimated by average speeds over shrinking intervals, and the estimates closed in on one number. A graph settling onto a horizontal asymptote and the sum of an infinite geometric series are numbers that outputs approach without reaching them in one step. The limit is the single idea under all of these: the height a function heads toward as its input heads toward a target.

The lesson takes two days. Part 1 is the idea of a limit, one-sided limits, and how a limit differs from the value of the function; Examples 1 to 4 go with it. Part 2 is limits that do not exist, limits as x \to \infty, and the infinite sum as a limit; Examples 5 to 7 go with it.

The procedure

Part 1: the idea of a limit

Definition. Let f be defined at every input near a on both sides, except perhaps at a itself. Suppose that for every input x \ne a close enough to a, the output f(x) is as close as you like to one number L. Then L is the limit of f(x) as x approaches a, written

Three things in the definition matter. The input approaches a from both sides. The input never equals a, so the value f(a), if there is one, plays no part. And “as close as you like” means any closeness at all can be demanded, and it is met once x is near enough to a. Making that sentence exact, with a named tolerance and a window of inputs, is the epsilon delta definition met in calculus.

Reading a limit from a graph. Put a finger on the graph a little to the left of x = a and slide it toward a along the curve. Do the same from the right. If both fingers head toward the same point (a, L), the limit is L. Whatever sits directly above a (a closed dot, an open circle, a dot at another height, or nothing) is ignored, because the definition never looks at x = a.

One-sided limits. Often the two sides need to be watched separately. Write

when the outputs approach L as x approaches a through inputs less than a, from the left; \lim_{x \to a^+} f(x) is the same from the right. This is the notation of Continuous graphs, jumps and holes and Vertical asymptotes. The two-sided limit asks both sides to close in on one number, so the limit \lim_{x \to a} f(x) exists exactly when both one-sided limits exist and are equal, and then it is their common value. This is One sided limits and when a limit exists.

Limit versus value. Figure 1 shows the same curve three ways at x = a.

In A the graph passes through (a, L), so the limit and the value are both L. In B the point is missing, a hole: the limit is L but f(a) is undefined. Holes appear in rational functions where a factor cancels, as in Graphing rational functions. In C the point has been moved: f(a) is another number, and the limit is still L. The limit and the value answer different questions. When they agree, as in A, the function is continuous at a; that is the definition of Continuity, met informally in Continuous graphs, jumps and holes.

Estimating a limit from a table. With a formula and no graph, evaluate f at inputs closing in on a from both sides, such as a - 0.1, a - 0.01, a - 0.001 and a + 0.001, a + 0.01, a + 0.1. If the outputs on both sides settle on the same digits, those digits estimate the limit. A table is evidence, not proof: it checks a few inputs, and the definition speaks of all of them near a. Example 6 shows a table that looks convincing and is wrong. When the formula can be simplified, for instance by canceling a factor, the algebra confirms the estimate; that is Limits by algebra: factoring and rationalizing.

Part 2: limits that do not exist, and limits at infinity

A limit fails to exist when the outputs near a do not close in on one number. There are three ways this happens.

• A jump. Both one-sided limits exist but are different numbers, so no single L serves both sides. See Discontinuities and how to remove them.
• Unbounded growth. Near a vertical asymptote the outputs pass every number. Writing \lim_{x \to a^+} f(x) = \infty records how the limit fails: the outputs grow without bound through positive values. The symbol \infty is not a number, so the limit still does not exist. If both sides go to \infty, write \lim_{x \to a} f(x) = \infty. This is Infinite limits and vertical asymptotes.
• Endless oscillation. However close x comes to a, the outputs keep swinging between heights that stay apart, as \displaystyle \sin\left(\frac{1}{x}\right) does near 0.

Limits at infinity. \lim_{x \to \infty} f(x) = L means that the outputs are as close as you like to L for every large enough x. On a graph the curve settles onto the horizontal line y = L far to the right, a horizontal asymptote. The limit as x \to -\infty is the same far to the left. A table uses inputs such as 10, 100, 1000.

An infinite sum as a limit. A sequence is a function of n = 1, 2, 3, \ldots, so the same idea applies as n \to \infty. The partial sums S_n of a series form a sequence, and the sum of the series is

when that limit exists. This is the definition of the sum in Infinite geometric series, written in limit notation. For a geometric series with |r| < 1, \displaystyle S_n = \frac{a_1(1 - r^n)}{1 - r} and r^n approaches 0, so \displaystyle S = \frac{a_1}{1 - r}. When |r| \ge 1 and a_1 \ne 0 the partial sums have no limit and the series has no sum.

Worked examples

Boundaries

• The two-sided limit needs f defined on both sides near a. Where inputs exist on one side only, as for \displaystyle \sqrt{x} at 0, only the one-sided limit is asked: \displaystyle \lim_{x \to 0^+} \sqrt{x} = 0.
• A limit, when it exists, is one number. “\lim_{x \to a} f(x) = \infty” is not a limit; it says how the limit fails.
• The limit at a says nothing about f(a), and f(a) says nothing about the limit.
• A table never proves a limit (Example 6). A computer graph can hide a hole, since one missing point has no width on the screen, and cannot draw an oscillation like Figure 4 faithfully near 0.
• A graph may cross its horizontal asymptote and still have \lim_{x \to \infty} f(x) = L; only the far-out behavior counts.

Where this goes next

In calculus limits are found exactly rather than estimated. The limit laws say limits respect sums and products, Limits by algebra: factoring and rationalizing removes the \displaystyle \frac{0}{0} of Example 2, and The squeeze theorem and the limit of sine x over x settles a limit no algebra reaches. Continuity is defined as “the limit equals the value”. The shrinking intervals of Rates at an instant from shrinking intervals become the derivative, a limit of average rates, and areas become the limit of sums of rectangles. Far-out and blow-up behavior return as Limits at infinity and horizontal asymptotes and Infinite limits and vertical asymptotes. Finally, The epsilon delta definition of a limit makes “as close as you like” exact.

Exercises

Part 1: reading the graph of k

1. Find \lim_{x \to -3} k(x) and k(-3).
2. Find \lim_{x \to -1^-} k(x), \lim_{x \to -1^+} k(x), \lim_{x \to -1} k(x) and k(-1).
3. Find \lim_{x \to 1} k(x) and k(1).

Part 1: tables and formulas

4. For \displaystyle f(x) = \frac{x^2 + 2x - 8}{x - 2}, find f(1.9), f(1.99), f(2.01) and f(2.1), estimate \lim_{x \to 2} f(x), and confirm it by factoring.
5. Find \displaystyle \lim_{x \to -5} \frac{x^2 - 25}{x + 5}, using the inputs -5.1 and -4.9 to check.
6. Estimate \displaystyle \lim_{x \to 0} \frac{\sqrt{x + 9} - 3}{x} from its values at x = \pm 0.01 and x = \pm 0.001, to five decimal places.
7. Estimate \displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x^2} (x in radians) from its values at x = 0.1 and x = 0.01, to six decimal places. Why do the negative inputs give nothing new?
8. Let q(x) = 2x + 1 for x < 3, q(3) = 4, and q(x) = 10 - x for x > 3. Find both one-sided limits at 3, the limit, and q(3).
9. Let \displaystyle r(x) = \frac{x^2 - 1}{x - 1} for x \ne 1 and r(1) = 5. Find \lim_{x \to 1} r(x) and r(1).

Part 2: reading the graph of k

10. Find \lim_{x \to 3^-} k(x), \lim_{x \to 3^+} k(x), \lim_{x \to 3} k(x) and k(3).
11. Describe k(x) as x \to 5^- and as x \to 5^+. Does \lim_{x \to 5} k(x) exist?
12. Find \lim_{x \to \infty} k(x), and check with the formula at x = 105.

Part 2: formulas and tables

13. Let \displaystyle j(x) = \frac{|x - 4|}{x - 4}. Find \lim_{x \to 4^-} j(x), \lim_{x \to 4^+} j(x) and \lim_{x \to 4} j(x).
14. Describe \displaystyle \frac{1}{(x + 1)^2} as x \to -1, using the inputs -1.01 and -0.99.
15. Evaluate \displaystyle \frac{5x - 2}{x + 1} at x = 10, 100 and 1000 (five decimal places) and estimate its limit as x \to \infty.
16. Find \lim_{x \to -\infty} 2^x, and describe 2^x as x \to \infty.
17. Show that \displaystyle \lim_{x \to 0} \cos\left(\frac{1}{x}\right) does not exist by evaluating at \displaystyle x = \frac{1}{2m\pi} and at \displaystyle x = \frac{1}{(2m + 1)\pi} for positive integers m.

Part 2: infinite sums

18. For \displaystyle 9 + 6 + 4 + \frac{8}{3} + \cdots, find S_1 to S_4 and \lim_{n \to \infty} S_n.
19. The repeating decimal 0.272727\ldots is the limit of the partial sums of 0.27 + 0.0027 + 0.000027 + \cdots. Find it as a fraction.
20. Find the partial sums of 2 - 2 + 2 - 2 + \cdots. Do they have a limit?

Reasoning

21. Primitivo says “f(3) = 7, so \lim_{x \to 3} f(x) = 7.” Inocencia says “\lim_{x \to 3} f(x) = 7, so f(3) = 7.” Describe a graph that shows each claim is false.
22. Explain why a function cannot have \lim_{x \to a} f(x) = 3 and \lim_{x \to a} f(x) = 4 at once.
23. Gumersinda’s table of some f at x = 0.1, 0.01, 0.001 shows 2.1, 2.01, 2.001. Give two reasons this alone does not show \lim_{x \to 0} f(x) = 2.
24. Use partial sums to explain why a_1 + a_1 + a_1 + \cdots (ratio r = 1, a_1 \ne 0) has no sum.