What a Table of Values Is Telling You
A limit asks what value a function is approaching as the input approaches a particular number. A table of values gives you numerical evidence for that behavior.
Suppose the question asks you to estimate
You do not need to know what happens exactly at . Instead, look at what happens to when gets closer and closer to from both sides.
The key idea is simple:
If the function values from the left and the right both appear to approach the same number, that number is your estimate of the limit.
Reading the Table Correctly
The most important skill is recognizing which inputs are approaching the target from the left and which are approaching from the right.
Values of that are less than the target value. For example, approaches 2 from the left.
Values of that are greater than the target value. For example, approaches 2 from the right.
For a two-sided limit, you need evidence from both sides. This is one of the most important habits to develop for AP Calculus.
Worked Example 1: A Two-Sided Limit
Use the table to estimate .
| 1.9 | 4.61 |
| 1.99 | 4.9601 |
| 1.999 | 4.996001 |
| 2.001 | 5.004001 |
| 2.01 | 5.0401 |
| 2.1 | 5.41 |
The "From Both Sides" Rule
For a two-sided limit to exist, the left-hand and right-hand limits must agree.
In table problems, this means you should deliberately inspect entries on both sides of the target input. Do not simply find the closest value in the table and stop.
How Close Is Close Enough?
Tables rarely give you an input of exactly . Instead, they usually provide values such as and .
The closer the inputs get to the target, the more useful the corresponding function values become for estimating the limit.
For example, suppose you see:
| 2.9 | 7.8 |
| 2.99 | 7.98 |
| 2.999 | 7.998 |
| 3.001 | 8.002 |
| 3.01 | 8.02 |
| 3.1 | 8.2 |
The values closest to suggest that is approaching . The entries farther away are useful for seeing the overall pattern, but the closest entries provide the strongest evidence.
Worked Example 2: The Function Value Can Be Different
Use the table to estimate .
| 0.9 | 2.89 |
| 0.99 | 2.9899 |
| 0.999 | 2.998999 |
| 1 | 7 |
| 1.001 | 3.001001 |
| 1.01 | 3.0101 |
| 1.1 | 3.11 |
This is a central idea in calculus: a limit describes nearby behavior, not necessarily the value at the point itself.
If the table contains a value for , do not assume that it is the answer. A limit looks at what happens as approaches , not necessarily what happens when equals .
Worked Example 3: A One-Sided Limit
Use the table to estimate .
| 2.1 | 4.2 |
| 2.01 | 4.02 |
| 2.001 | 4.002 |
| 1.999 | -4.001 |
| 1.99 | -4.01 |
Notice that the values on the left side of 2 approach a completely different number. That matters for the two-sided limit, but it does not change the right-hand limit.
Worked Example 4: When the Two-Sided Limit Does Not Exist
Use the table to determine whether exists.
| 4.9 | 1.01 |
| 4.99 | 1.001 |
| 4.999 | 1.0001 |
| 5.001 | 2.9999 |
| 5.01 | 2.999 |
| 5.1 | 2.99 |
A Reliable Four-Step Method
- Find the target value. Identify the number that is approaching.
- Separate the two sides. Values below the target approach from the left; values above it approach from the right.
- Look at the closest inputs. Use the function values associated with inputs closest to the target.
- Compare the two sides. If both sides approach the same number, that is your estimated two-sided limit. If they approach different numbers, the two-sided limit does not exist.
What If the Values Become Very Large?
Tables can also provide evidence that a function is growing without bound. For example:
| 1.1 | 10 |
| 1.01 | 100 |
| 1.001 | 1,000 |
| 1.0001 | 10,000 |
| 1.00001 | 100,000 |
As approaches 1 from the right, the function values become arbitrarily large. This provides evidence for
Be careful: infinity is not a number. Writing a limit as infinity describes unbounded behavior rather than saying that the function approaches a finite value.
Common Mistakes
For a two-sided limit, you need information from both sides of the target. A value that looks convincing from the left is not enough.
The value of may be completely different from the limit. The limit is determined by nearby values as approaches the target.
A table may contain several values that show the general trend, but the entries closest to the target usually provide the best estimate of the limit.
The notation means approach from the left, while means approach from the right. The ordinary limit requires both sides to agree.
AP Exam Strategy
When an AP Calculus question gives you a table and asks for a limit, don't overcomplicate it. The question is usually testing whether you can interpret the numerical behavior correctly.
- Circle or mentally identify the target input.
- Find the closest input values on either side of it.
- Read the corresponding function values.
- Determine what those values are approaching.
- Check whether the left and right sides agree.
- Report the requested limit with appropriate precision.
If the question asks you to estimate a limit, you generally do not need an elaborate algebraic derivation. The table itself is the evidence.
Practice Problems
Use the table to estimate .
| 2.9 | 5.79 |
| 2.99 | 5.9799 |
| 2.999 | 5.997999 |
| 3.001 | 6.002001 |
| 3.01 | 6.0201 |
| 3.1 | 6.21 |
Show answer
Use the table to estimate .
| 0.9 | -1.1 |
| 0.99 | -1.01 |
| 0.999 | -1.001 |
| 1.001 | -0.999 |
| 1.01 | -0.99 |
| 1.1 | -0.9 |
Show answer
The table below gives values near . Determine whether the two-sided limit exists.
| 1.99 | 4.01 |
| 1.999 | 4.001 |
| 2.001 | 6.001 |
| 2.01 | 6.01 |
Show answer
Use the table to estimate .
| 4.1 | 8.2 |
| 4.01 | 8.02 |
| 4.001 | 8.002 |
| 3.999 | 7.999 |
| 3.99 | 7.99 |
Show answer
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