Learning Objectives
- Estimate a limit using numerical tables of function values.
- Recognize convergence (approaching a single value) versus divergence.
- Use tables to support graphical and algebraic reasoning.
Motivation
When a graph is complicated or unavailable, we can build a table of values around the point of interest. If the outputs settle toward a number, that’s the estimated limit.
How to Estimate from Tables
- Pick values of \(x\) close to the target \(a\), both smaller and larger.
- Compute \(f(x)\) for each chosen \(x\).
- Observe whether the outputs approach a single number.
- If they do, that’s the limit. If left and right differ, the limit DNE.
Example 1: \( \frac{x^2-1}{x-1} \) as \(x\to 1\)
| \(x\) | \(f(x)\) | \(x\) | \(f(x)\) |
|---|---|---|---|
| 0.9 | 1.9 | 1.1 | 2.1 |
| 0.99 | 1.99 | 1.01 | 2.01 |
| 0.999 | 1.999 | 1.001 | 2.001 |
As values of \(x\) get closer to 1, \(f(x)\) approaches 2. So \( \lim_{x\to 1} \frac{x^2-1}{x-1} = 2 \).
Example 2: Oscillating Table
For \( f(x) = \sin\!\left(\tfrac{1}{x}\right) \) as \(x\to 0\):
| \(x\) | \(f(x)\) |
|---|---|
| 0.1 | \(\sin(10)\approx -0.54\) |
| 0.01 | \(\sin(100)\approx -0.51\) |
| 0.001 | \(\sin(1000)\approx 0.83\) |
The values jump around without settling → the limit does not exist.
Interactive Table Builder
Adjust \(a\) and \(h\). The table shows values of \(f(x)=\frac{x^2-1}{x-1}\) as \(x\) approaches \(a=1\) from both sides.
0.10
As \(h\to 0\), the table values approach 2, even though \(f(1)\) is undefined.
Quick Reference
- Tables show numerical evidence of a limit.
- Choose \(x\) close to \(a\) from both sides.
- If outputs approach the same value, that’s the limit.
- If outputs oscillate or differ, the limit DNE.
Try It
- Estimate \( \lim_{x\to 2} \frac{x^2-4}{x-2} \) using a table.
- Make a table for \( \frac{\sin x}{x} \) as \(x\to 0\). What value do you suspect?
- Use a table to explore \( \lim_{x\to 0} \sin\!\left(\tfrac{1}{x}\right) \). Why does it not exist?
Key Takeaways
- Tables provide a numerical way to approximate limits.
- Approach from both left and right sides.
- Look for convergence — if values settle on a single number, that’s the limit.
- If values diverge or oscillate, the limit does not exist.