Estimating Limits from Tables

Using Tables to Find Limits

One of the most intuitive ways to understand limits is by creating a table of values and observing the pattern.

The Strategy

To estimate $\lim_{x \to a} f(x)$:

1. Choose values approaching from the left (x < a)
2. Choose values approaching from the right (x > a)
3. Calculate f(x) for each value
4. Look for a pattern - what value are the outputs approaching?

Example Setup

Let's find $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ using a table.

Note: We can't just plug in x = 2 because we'd get $\frac{0}{0}$ (undefined!)

From the left (x < 2):

| x | f(x) = $\frac{x^2 - 4}{x - 2}$ | |---|---| | 1.9 | $\frac{(1.9)^2 - 4}{1.9 - 2} = \frac{-0.39}{-0.1} = 3.9$ | | 1.99 | $\frac{(1.99)^2 - 4}{1.99 - 2} = \frac{-0.0399}{-0.01} = 3.99$ | | 1.999 | $\frac{(1.999)^2 - 4}{1.999 - 2} \approx 3.999$ |

From the right (x > 2):

| x | f(x) = $\frac{x^2 - 4}{x - 2}$ | |---|---| | 2.1 | $\frac{(2.1)^2 - 4}{2.1 - 2} = \frac{0.41}{0.1} = 4.1$ | | 2.01 | $\frac{(2.01)^2 - 4}{2.01 - 2} = \frac{0.0401}{0.01} = 4.01$ | | 2.001 | $\frac{(2.001)^2 - 4}{2.001 - 2} \approx 4.001$ |

The Pattern

Both sides are approaching 4!

Therefore: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$

Tips for Making Tables

1. Start reasonably close - Use values like 0.1, 0.01 away from a
2. Get progressively closer - Move to 0.01, 0.001, etc.
3. Check both sides - Always examine left and right approaches
4. Use a calculator - For complex functions, don't do it by hand
5. Look for agreement - If both sides approach the same value, that's your limit!

When Tables Fail

Tables are estimates! They might be misleading if:

• You don't get close enough to the point
• The function oscillates wildly near the point
• There's a very steep change right at the point

Tables are excellent for building intuition, but algebraic methods (coming in later lessons) are more precise.