Short on time? This page is the recap. For explanations, the diagram and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (x → a) f(x) is the limit as x approaches a; x → a⁻ means from the left (x < a); x → a⁺ means from the right (x > a).
Recap
- A table gives numerical evidence about a limit, just as a graph gives visual evidence.
- Read the rows with x < c moving towards c: that estimates the left-hand limit.
- Read the rows with x > c moving towards c: that estimates the right-hand limit.
- If both sides approach the same number L, estimate the limit as L. If they differ, the two-sided limit does not exist.
- Ignore f(c) even if the table shows it.
- Round your estimate to the decimal places on which the closest values from both sides agree.
Key relationships
| What you see in the table | Conclusion |
|---|---|
| Both sides approach the same number L | lim (x → c) f(x) ≈ L |
| Left approaches L₁, right approaches L₂, with L₁ ≠ L₂ | One-sided limits differ; the limit does not exist |
| Values grow without bound (100, 10 000, 1 000 000, …) | Unbounded; no finite limit (you may write = ∞ or = −∞ when both sides share one sign) |
| Values keep jumping between numbers however close x is | Oscillation; the limit does not exist |
| f(c) differs from the trend | Irrelevant to the limit; the limit follows the trend |
Good x values: c ± 0.1, c ± 0.01, c ± 0.001, then c ± 0.0001 if you need more precision.
Assumptions behind the method
- The pattern you see continues for every x between your sampled values and c. A table cannot guarantee this.
- The calculator values are accurate. Extremely close x values (such as c + 10⁻¹⁵) can give rounding nonsense.
- Trig work is done in radians.
Mistakes to avoid
- Taking the closest table value as the exact limit (for example, answering 9.8821 instead of about 9.9).
- Using f(c) from the table as the limit.
- Averaging two different one-sided values.
- Reading one side only for a two-sided limit.
- Calling huge values “the limit” when they keep growing.
- Trusting a “perfect” pattern: sin(π/x) at x = 0.1, 0.01, 0.001 gives 0 each time, yet the limit does not exist.
Quick self-check
- Left values 4.9, 4.99, 4.999; right values 5.1, 5.01, 5.001. Estimate the limit. (5)
- The same table also shows f(c) = 8. Does that change your answer? (No. The limit is still about 5.)
- Left values approach 3 and right values approach 1. What is the two-sided limit? (It does not exist. It is not 2.)
- Closest values are 3.1372 (left) and 3.1461 (right). To how many decimal places can you trust an estimate? (1: both round to 3.1, but to 2 decimal places they give 3.14 and 3.15.)
Next: practice questions.