1.4 Estimating Limit Values from Tables

Using tables to estimate limits

When a function is given as a table of input-output values, you can estimate a limit by examining values of x that get closer and closer to the target value from both sides. Look at outputs as x approaches from the left and from the right — if both sides trend toward the same y-value, that value is your estimated limit.

Checking left and right behavior

A limit only exists at a point if the function approaches the same value from both directions. If table values from the left approach one number and values from the right approach a different number, the limit does not exist at that point, even if the table shows values close to the target x.

Limitations of table estimation

Tables only give approximations, not exact limit values, since we can't see behavior at every possible x. Closer-spaced x-values near the target generally give a more reliable estimate. Always be cautious: a table might miss a sudden jump, hole, or oscillation in the actual function between listed points.

Key terms

How do you estimate a limit from a table?

Check the output values as x approaches the target from both the left and right sides; if both trend toward the same number, that's the estimated limit.

When does a limit not exist based on table data?

When the left-side and right-side values approach different numbers.

Limitation of table estimation

Tables only show discrete points, so they might miss jumps, holes, or oscillations between listed values — giving only an approximation, not an exact limit.

Best x-values to use in a table

Values very close to the target x from both sides, since closer spacing gives a more accurate estimate.

Two-sided limit requirement

The function must approach the same value from both the left and right for the limit to exist at that point.

Try an AP-style question

What is the best estimate of lim(x→2) f(x)?

1. 3.996
2. 4
3. 2
4. 4.41