AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/1/1-4)
Unit 1 · Topic 1.4
1.4 Estimating Limit Values from Tables
A table of x-values that creep toward c from both sides gives numerical evidence of a limit. You look at the trend in the outputs from each side. A table can suggest a limit but can't prove one, and a badly chosen table can even mislead you.
Key terms
- table of values
- estimate
- approaching from the left
- approaching from the right
Setting up a table
Pick x-values closer and closer to c from the left, like 1.9, 1.99, 1.999 when c = 2, and from the right, like 2.1, 2.01, 2.001. Compute f(x) for each.
Read the left column toward c and the right column toward c. If both sets of outputs close in on the same number, that number is your estimate for the limit.
What the table can and can't tell you
Read the outputs from each side as the inputs close in on c. Here is what the patterns mean:
- If the left and right outputs approach the same number, the limit appears to be that number.
- If they approach two different numbers, the two-sided limit appears not to exist (a jump).
- If the outputs keep growing in size, the function appears to be unbounded near c.
- If the outputs bounce around, the function may be oscillating.
How precise is the estimate?
A table only shows finitely many points. Whatever happens between them is invisible. That's why the exam uses words like “estimate” or “appears” for table-based limits, and why algebra (1.5 and 1.6) is used to confirm.
If the outputs from both sides agree to two decimal places, you can trust the estimate to about that precision. Rounding the estimate to a clean value, like 12, is fine when the trend clearly points there.
When a table comes from real data (like temperatures at certain times), you usually can't zoom in further. Use what's there and say the limit is approximately the trend value.
Using a calculator table
Your graphing calculator's table feature can build these tables for you. Start close to c and use a small step, or type in x-values one at a time, like 2.01, 2.001 and 2.0001.
Don't push too far. With x-values extremely close to c, like c + 0.0000000001, the calculator has to subtract two nearly equal numbers and can lose accuracy, giving strange outputs. A few steps toward c is enough to see the trend.
Also watch for a calculator error at x = c itself. That just means f(c) is undefined, which says nothing about the limit.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Estimating a limit from both sides
Let f(x) = (x³ − 8) / (x − 2). Make a table using x = 1.9, 1.99, 1.999 and 2.001, 2.01, 2.1, and estimate lim (x→2) f(x).
Show the solutionHide the solution
- Step 1: Left side: f(1.9) = 11.41, f(1.99) = 11.9401, f(1.999) = 11.994001.
- Step 2: Right side: f(2.1) = 12.61, f(2.01) = 12.0601, f(2.001) = 12.006001.
- Step 3: Both sides are closing in on 12: the left values creep up toward it and the right values creep down toward it.
- Step 4: Algebra confirms it: x³ − 8 = (x − 2)(x² + 2x + 4), so for x ≠ 2, f(x) = x² + 2x + 4, which approaches 4 + 4 + 4 = 12.
Answer: lim (x→2) f(x) appears to be 12 (and algebra confirms it equals 12).
- Example 2Calculator allowed
Trap: a table that lies
A student evaluates f(x) = sin(π/x) at x = 0.1, 0.01 and 0.001 and gets 0 every time. They conclude lim (x→0⁺) f(x) = 0. What went wrong?
Show the solutionHide the solution
- Step 1: Check the values: π/0.1 = 10π, π/0.01 = 100π and π/0.001 = 1000π. Sine of any whole multiple of π is 0, so the outputs really are 0.
- Step 2: But try x = 0.3, 0.03, 0.003. Then π/x = 10π/3, 100π/3, 1000π/3, and each sine is −√3/2 ≈ −0.866.
- Step 3: So near 0 the function takes the value 0 and also the value −0.866, again and again. In fact, it swings between −1 and 1 infinitely often.
- Step 4: The student's x-values happened to land exactly where the function is 0. The outputs don't settle on one number.
Answer: The limit does not exist. The table sampled only special points; sin(π/x) oscillates between −1 and 1 as x→0⁺.
Common mistakes
- Only checking one side. A table that goes 1.9, 1.99, 1.999 shows the left-hand limit only.
- Treating the value at x = c in a table as the limit. If the table includes f(2), ignore it when estimating lim (x→2) f(x).
- Claiming certainty from a table. Say the limit “appears to be” the value unless you also back it up with algebra.
On the exam
- Table questions are usually multiple choice: you pick the limit the table suggests, or the statement the table supports. Read both directions toward c.
- Data tables in free response (like those in later units) only give a few points, so an estimate is all you can give. That's expected.
Connected topics
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Check yourself
4 questions on 1.4 Estimating Limit Values from Tables. Pick an answer to see if you got it, and why.
Let f(x) = (sin x − x)/x³. Using a calculator to evaluate f at values of x close to 0, which of the following is the best estimate of lim (x→0) f(x) ?
| x | f(x) | g(x) |
|---|---|---|
| 0.9 | 2.71 | 1.9 |
| 0.99 | 2.9701 | 1.99 |
| 0.999 | 2.997 | 1.999 |
| 1.001 | 3.003 | −1.002 |
| 1.01 | 3.0301 | −1.0201 |
| 1.1 | 3.31 | −1.21 |
Selected values of f and g near x = 1
The table gives selected values of the functions f and g for x near 1. Based on the table, which of the following is the best estimate of lim (x→1) f(x) ?
Based on the table, which of the following statements about g is best supported?
Based on the table, which of the following is the best estimate of lim (x→1⁻) [f(x) · g(x)] ?
0 of 4 answered