Grade by Grade
A. Rate of Change · Limits and Rates of Change

The Limit of a Function: Graphs and Tables

A1.4About 70 minutes
Learning goals
  • I can read and write limit notation, including one-sided limits.
  • I can find limits from a graph, including at holes and jumps.
  • I can estimate a limit numerically with a table of values from both sides.
  • I can explain the difference between lim (x → a) f(x) and f(a).
  • I can decide whether a limit exists by comparing the left-hand and right-hand limits.

Words to know

limit of a function
lim (x → a) f(x) = L means f(x) gets as close to L as we like when x is close enough to a (from both sides), with x ≠ a.
left-hand limit
lim (x → a⁻) f(x): the value f(x) approaches as x approaches a from values less than a.
right-hand limit
lim (x → a⁺) f(x): the value f(x) approaches as x approaches a from values greater than a.
hole
A single point missing from a graph, shown with an open dot.
piecewise function
A function defined by different formulas on different parts of its domain.
does not exist (DNE)
A limit does not exist if the one-sided limits differ, if f(x) grows without bound, or if f(x) keeps oscillating.

From sequences to functions

In the last lesson, n marched off to infinity one whole number at a time. Now x will slide continuously toward a number a, from both sides, and we watch what f(x) does.

lim (x → a) f(x) = L

is read "the limit of f(x) as x approaches a is L". It means: when x is close enough to a (on either side, but not equal to a), f(x) is as close to L as we like.

The big surprise: the value f(a) does not matter. The limit is about the values near a. f(a) might equal L, might be a different number, or might not exist at all.

One-sided limits

Sometimes f behaves differently on the two sides of a.

  • lim (x → a⁻) f(x) is the left-hand limit: x approaches a from the left (x < a).
  • lim (x → a⁺) f(x) is the right-hand limit: x approaches a from the right (x > a).

The little − and + are not signs of numbers. They tell you which side you are coming from.

The two-sided limit exists exactly when both one-sided limits exist and are equal. Then lim (x → a) f(x) is that common value.

−1012345012345xyhole (1, 2)g(1) = 3open (3, 4)g(3) = 2y = g(x)At x = 1:lim g(x) = 2 (from both sides)g(1) = 3 (a different number!)At x = 3:left-hand limit = 4right-hand limit = 2so lim g(x) does not existOpen dot: not on the graph.Filled dot: the actual value.
The graph of g. At x = 1 there is a hole at height 2 and the actual value g(1) = 3 sits above it. At x = 3 the graph jumps from height 4 (open dot) down to height 2 (filled dot).
Worked example

Example 1: Reading limits from a graph

Use the graph of g above. Find (a) lim (x → 1) g(x) and g(1), (b) the one-sided limits at x = 3 and lim (x → 3) g(x), (c) g(3), (d) lim (x → 0) g(x) and lim (x → 4) g(x).

  1. (a) Trace the graph toward x = 1 from both sides: the heights approach 2 (the hole). So lim (x → 1) g(x) = 2. The filled dot says g(1) = 3.
  2. (b) From the left, the heights approach the open dot at 4: lim (x → 3⁻) g(x) = 4. From the right, they approach 2: lim (x → 3⁺) g(x) = 2.
  3. The one-sided limits are different, so lim (x → 3) g(x) does not exist.
  4. (c) The filled dot at x = 3 is at height 2, so g(3) = 2.
  5. (d) Near x = 0 the graph is the unbroken line y = x + 1, so the limit is 1. Near x = 4 it is y = 5 − x, so the limit is 1.

lim (x → 1) g = 2 but g(1) = 3; at x = 3 the limit DNE (left 4, right 2) and g(3) = 2; lim (x → 0) g = 1 and lim (x → 4) g = 1.

Read limits with your finger

To find a limit from a graph, put your finger on the graph to the left of x = a and slide it toward a. Note the height it approaches. Do the same from the right. Only after that, look at the dot exactly at x = a to find f(a). Keeping the two questions separate prevents most mistakes.

Worked example

Example 2: A limit from a table

Estimate lim (x → 0) (sin x)/x, with x in radians.

  1. The function is undefined at x = 0 (it would be 0/0), so we cannot just substitute.
  2. Make a table from both sides (sin x/x gives the same value at x and −x): x = ±0.1 gives 0.998 334; x = ±0.01 gives 0.999 983; x = ±0.001 gives 0.999 999 8.
  3. From both sides the values approach 1.

lim (x → 0) (sin x)/x = 1. This limit is the key to the derivative of sin x in the sibling unit Derivatives of Exponential and Sinusoidal Functions.

xy−3π−2π−ππ2π3π(0, 1) is a holey = (sin x)/xis undefined at x = 0,but as x → 0 fromeither side, y → 1.lim (sin x)/x = 1x→0(x in radians)
The graph of y = (sin x)/x has a hole at (0, 1). The function has no value at 0, but its limit there is 1.
Worked example

Example 3: A limit that is not a nice number

Estimate lim (x → 0) (2ˣ − 1)/x to three decimal places.

  1. Right side: x = 0.1 → 0.717 73; x = 0.01 → 0.695 56; x = 0.001 → 0.693 39.
  2. Left side: x = −0.1 → 0.669 67; x = −0.01 → 0.690 75; x = −0.001 → 0.692 91.
  3. The two sides squeeze together. They agree to 0.693 (to three decimal places).

The limit is about 0.693. (It is the natural logarithm of 2, ln 2, as you will see in Derivatives of Exponential and Sinusoidal Functions. Notice it is also the slope of the tangent to y = 2ˣ at x = 0.)

Worked example

Example 4: When one-sided limits disagree

Let f(x) = |x|/x. Find lim (x → 0⁻) f(x), lim (x → 0⁺) f(x) and lim (x → 0) f(x).

  1. For x > 0, |x| = x, so f(x) = x/x = 1. The right-hand limit is 1.
  2. For x < 0, |x| = −x, so f(x) = −x/x = −1. The left-hand limit is −1.
  3. The one-sided limits are different.

lim (x → 0⁻) f(x) = −1, lim (x → 0⁺) f(x) = 1, so lim (x → 0) f(x) does not exist. (f(0) is undefined too, but that is a separate fact.)

Three ways a limit can fail to exist

  1. Jump: the one-sided limits are different numbers (Examples 1 and 4).
  2. Unbounded: f(x) grows without bound near a, like 1/x² near 0. The next lesson, Limits Involving Infinity and Asymptotes, deals with these.
  3. Oscillation: f(x) keeps swinging between values, like sin(1/x) near 0 (see the Challenge).
Just for fun

A limit is like a cat that walks right up to your hand, sniffs it, and refuses to be touched. It gets as close as you like. Whether it actually arrives is none of the limit's business.

Hands-on activity

Sneaking up on a hole

You need: a calculator (in radian mode), grid paper and a pencil.

  1. Make a table of y = (x² − 1)/(x − 1) for x = 0.9, 0.99, 0.999 and x = 1.1, 1.01, 1.001.
  2. Try x = 1. What does your calculator say? Why?
  3. Plot your points and a few more (x = 0, 2, 3, −1) on grid paper. Draw the graph, and show what happens at x = 1 with an open dot.
  4. Repeat steps 1 and 2 for y = (1 − cos x)/x² near x = 0, using x = ±0.1, ±0.01, ±0.001.
  5. Write each limit in limit notation.

What to look for: the first table gives 1.9, 1.99, 1.999 and 2.1, 2.01, 2.001, so the limit is 2. At x = 1 the calculator gives an error (0 ÷ 0). The graph is the line y = x + 1 with a hole at (1, 2). The second table gives about 0.4996, 0.499 996, … from both sides, so the limit is 0.5.

Your turn

1

Use the graph of g. What is lim (x → 1) g(x)?

2

Use the graph of g. What is g(1)?

3

Use the graph of g. Find lim (x → 3⁻) g(x) and lim (x → 3⁺) g(x).

4

Use the graph of g. Does lim (x → 3) g(x) exist? Explain.

5

Estimate lim (x → 1) (x² − 1)/(x − 1) using x = 0.9, 0.99, 1.01 and 1.1.

6

f(x) = x + 2 for x < 0 and f(x) = x² for x ≥ 0. Find the one-sided limits at 0 and decide whether lim (x → 0) f(x) exists.

7

Estimate lim (x → 0) (1 − cos x)/x² (radians) using x = ±0.1 and ±0.01.

More practice: warm-up

1

Write "lim (x → 4) f(x) = 7" in words.

2

Find lim (x → 2) (3x + 1).

3

The left-hand and right-hand limits at x = 5 are both 8. What is lim (x → 5) f(x)?

4

The left-hand limit at x = 1 is 2 and the right-hand limit is 3. What is lim (x → 1) f(x)?

5

What is lim (x → a) c, where c is a constant?

More practice: on your own

1

f(x) = 2x + 1 for x < 2 and f(x) = x² + 1 for x ≥ 2. Find lim (x → 2) f(x) and f(2).

2

Estimate lim (x → 2) (x³ − 8)/(x − 2) using x = 1.99 and 2.01.

3

Describe a function with lim (x → 2) f(x) = 3 but f(2) = 1.

4

The floor function ⌊x⌋ gives the greatest integer less than or equal to x (for example, ⌊2.7⌋ = 2). Find the one-sided limits at x = 2, and lim (x → 2.5) ⌊x⌋.

5

Estimate lim (x → 0) (3ˣ − 1)/x to three decimal places.

Challenge

1

Explain why lim (x → 0) sin(1/x) does not exist.

2

f(x) = kx + 3 for x < 1 and f(x) = x² + 2k for x > 1. Find k so that lim (x → 1) f(x) exists, and give the limit.

3

Estimate lim (x → 0) (1 + x)^(1/x) with x = ±0.01 and ±0.001. Which number from the last lesson appears?

Check your understanding

Quiz

1.

What does lim (x → a) f(x) = L describe?

2.

If lim (x → 2⁻) f(x) = 6 and lim (x → 2⁺) f(x) = 6, and f(2) = 1, then lim (x → 2) f(x) is:

3.

For f(x) = |x|/x, what is lim (x → 0⁺) f(x)?

4.

A table gives f(0.9) = 2.71, f(0.99) = 2.97, f(1.01) = 3.03, f(1.1) = 3.31. The best estimate of lim (x → 1) f(x) is:

5.

What is lim (x → 0) (sin x)/x (x in radians)?

6.

Which situation guarantees that lim (x → a) f(x) does not exist?

For parents and teachers

Big idea (A1.4): lim (x → a) f(x) describes what f(x) approaches near a, not what happens at a. A two-sided limit exists only when the left-hand and right-hand limits agree.

How to help: Sketch a graph with a hole and a jump (like the graph of g). Point to x-values and ask, "What does the graph approach from the left? From the right? What is the actual value?" Then switch roles and let your teen quiz you.

Common difficulties:

  • Answering with f(a) instead of the limit, especially at a hole.
  • Reading a⁻ and a⁺ as negative and positive numbers instead of directions.
  • Checking only one side in a table.
  • Using degree mode for trigonometric limits; (sin x)/x → 1 only in radians.