Combining Graphs and Tables
A graph shows the overall path a function follows near a target, while a table gives numerical samples along that path. Used together, these representations can support a clearer estimate of a limit than either one alone. The central task is to track what the function values approach as the inputs approach a target—not simply to read the height of a point drawn at that input.
For a target \(x=a\), approaching from the left means using inputs less than \(a\) and moving closer to \(a\). Approaching from the right means using inputs greater than \(a\) and moving closer. The function may have a value at \(a\), a different value, or no value there; none of these possibilities by itself determines the limiting behavior nearby.
A Reliable Reading Process
For a graph-and-table question, work systematically. First identify the target input and the direction of approach. Then trace the graph toward the target without stopping at a filled point. In the table, sort the inputs into those below and above the target, and compare the corresponding outputs as each group gets closer. Finally, state the one-sided estimates and decide whether they agree.
Identify the input \(a\), and keep track of which values lie to its left and right.
Follow the curve toward \(x=a\) from each side and estimate the height each side approaches.
Compare outputs for inputs below \(a\) separately from outputs for inputs above \(a\).
If both sides support the same height, report that as the estimated two-sided limit. If they support different heights, the two-sided limit does not exist.
In these examples, graph and table readings provide evidence for an estimate. A finite set of plotted points or table entries cannot, by itself, prove the exact behavior for every input arbitrarily close to the target. The conclusion should match the evidence: say what is estimated, and distinguish it from a proof based on an exact rule.
Worked Example: A Hole in the Graph and a Different Function Value
Worked Example: A Hole in the Graph and a Different Function Value
A graph has a straight trace approaching an open circle at \((2,3)\) from both sides. It also has a filled point at \((2,1)\). A nearby table gives:
| Input \(x\) | Function value \(f(x)\) | Side of 2 |
|---|---|---|
| 1.90 | 2.80 | Left |
| 1.99 | 2.98 | Left |
| 2.01 | 3.02 | Right |
| 2.10 | 3.20 | Right |
Estimate the one-sided limits, the two-sided limit, and the function value at \(x=2\).
Step 1: Read the graph. From the left, the trace rises toward height \(3\). From the right, it approaches the same height \(3\). The open circle marks that approached height; it is not a filled point.
Step 2: Check the left-side table values. At \(x=1.90\), the output is \(2.80\), which is \(3-2.80=0.20\) below \(3\). At \(x=1.99\), it is \(2.98\), or \(3-2.98=0.02\) below \(3\). The closer left input has an output closer to \(3\).
Step 3: Check the right-side table values. At \(x=2.01\), the output is \(3.02\), or \(3.02-3=0.02\) above \(3\). At \(x=2.10\), it is \(3.20\), or \(3.20-3=0.20\) above \(3\). The closer right input also has an output closer to \(3\).
Both representations support the estimates
The filled point gives \(f(2)=1\). Thus the estimated limit is \(3\), while the function value is \(1\). The limit conclusion describes nearby outputs, not the filled point's height.
Check: The table's closest left and right inputs, \(1.99\) and \(2.01\), have outputs \(2.98\) and \(3.02\), each \(0.02\) from \(3\). That is consistent with the graph's shared approached height.
Why Both Sides Determine a Two-Sided Limit
A two-sided approach to \(a\) includes inputs on both sides of \(a\). Therefore, for the outputs to approach one height \(L\), they must approach \(L\) from the left and from the right. This gives a practical test for conclusions based on a graph or table.
The reason is that “near \(a\)” includes both sides. If both sides approach \(L\), then values close enough to \(a\), whether below or above it, have outputs close to \(L\). Conversely, if one side approaches a different height, or fails to approach the same height, then values from both sides cannot share a single limiting height. The two one-sided checks are not optional details; they describe the two parts of the approach.
Worked Example: Unequal One-Sided Estimates
Worked Example: Unequal One-Sided Estimates
A graph approaches height \(2\) as \(x\) approaches \(0\) from the left, and height \(-1\) from the right. A filled point is drawn at \((0,4)\). A table gives:
| Input \(x\) | Function value \(f(x)\) | Side of 0 |
|---|---|---|
| \(-0.10\) | 2.10 | Left |
| \(-0.01\) | 2.01 | Left |
| 0.01 | \(-0.99\) | Right |
| 0.10 | \(-0.90\) | Right |
Estimate the one-sided limits and decide whether the two-sided limit exists.
Step 1: Estimate from the left. The output \(2.10\) at \(x=-0.10\) is \(0.10\) above \(2\). At \(x=-0.01\), the output \(2.01\) is \(0.01\) above \(2\). The closer input supports an approach to \(2\), matching the graph.
Step 2: Estimate from the right. The output \(-0.99\) at \(x=0.01\) is \(0.01\) above \(-1\). At \(x=0.10\), the output \(-0.90\) is \(0.10\) above \(-1\). These values support an approach to \(-1\), also matching the graph.
The one-sided estimates are therefore
Since \(2\ne -1\), the one-sided estimates do not match, so there is no two-sided limit at \(x=0\). The filled point gives \(f(0)=4\), but that value does not change the different behavior on the two sides.
Check: The closest listed inputs on both sides are \(-0.01\) and \(0.01\); their outputs, \(2.01\) and \(-0.99\), remain close to different heights. This agrees with the graph and the conclusion.
Worked Example: Matching Evidence at a Target
Worked Example: Matching Evidence at a Target
A graph approaches an open circle at \((1,1.5)\) from both directions. A filled point is also shown at \((1,2.5)\). The table is:
| Input \(x\) | Function value \(f(x)\) | Side of 1 |
|---|---|---|
| 0.90 | 1.48 | Left |
| 0.99 | 1.498 | Left |
| 1.01 | 1.502 | Right |
| 1.10 | 1.52 | Right |
Estimate the two-sided limit and compare it with \(f(1)\).
Step 1: Compare left-side outputs to \(1.5\). At \(x=0.90\), \(1.5-1.48=0.02\). At \(x=0.99\), \(1.5-1.498=0.002\). As the inputs move closer to \(1\) from the left, the displayed outputs move closer to \(1.5\).
Step 2: Compare right-side outputs to \(1.5\). At \(x=1.01\), \(1.502-1.5=0.002\). At \(x=1.10\), \(1.52-1.5=0.02\). The closer right input again has an output closer to \(1.5\).
The graph and table support the estimate
The filled point gives \(f(1)=2.5\), which differs from the estimated limiting value by \(2.5-1.5=1\). The graph indicates that the nearby trace approaches \(1.5\), not the height of the filled point.
Check: The closest inputs, \(0.99\) and \(1.01\), have outputs \(1.498\) and \(1.502\), each \(0.002\) from \(1.5\). Their matched distances agree with the two graph traces approaching the same height.
What the Evidence Can Support
Graphs and tables are especially useful for estimating a limit and checking whether behavior from the left agrees with behavior from the right. They also help distinguish a limiting height from a function value at the target. However, a table contains only selected inputs, and a drawn graph has limited visual precision. Close values suggest a trend; they do not rule out every possible behavior between the displayed samples.
For that reason, report the conclusion carefully. If the graph and table agree, say they support an estimate of the limit. If the one-sided estimates differ, say that the evidence indicates no two-sided limit. Do not claim a general proof from a handful of entries alone. In each case, keep the three quantities conceptually separate: the left-hand limit, the right-hand limit, and the function value at the target.
Check Your Understanding
For each question, use the graph description and numerical evidence together, and state what the evidence justifies.
- A graph approaches height \(4\) from both sides of \(x=3\). The nearby outputs are \(3.9\) and \(4.1\). What two-sided limit is supported?
- As \(x\) approaches \(5\), the left-hand estimate is \(0\) and the right-hand estimate is \(2\). What conclusion follows about the two-sided limit?
- A graph has an open circle at \((1,6)\) and a filled point at \((1,2)\). What does each point indicate?
- Why should table entries below a target be compared separately from entries above it?
- Can a few matching table values prove an exact limit? Explain what they can support and what they cannot establish by themselves.
