Why Compare Representations?
A graph gives a visual picture of how a function behaves near a target input. A table gives selected numerical samples of that behavior. Either representation can help estimate a limit, but a two-sided limit depends on what happens on both sides of the target.
When a graph and a table provide information about different sides, comparing them can make the evidence clearer. The key question is whether the left-hand behavior and the right-hand behavior approach the same output value.
What Counts as Left-Hand and Right-Hand Evidence?
Approaching \(a\) from the left means using inputs less than \(a\) that get closer to \(a\). Approaching from the right means using inputs greater than \(a\) that get closer to \(a\). In a table, the input values show which side is being sampled. On a graph, follow the curve toward \(x=a\) along the corresponding side.
The left-hand limit is written \(\lim_{x\to a^-}f(x)\); the right-hand limit is written \(\lim_{x\to a^+}f(x)\). If both exist and are equal to the same number \(L\), then the two-sided limit exists and equals \(L\):
The arrows describe the direction of approach in the input, not whether the outputs increase or decrease. A branch can approach a value while rising or while falling; what matters is the output value it approaches.
How the One-Sided Comparison Works
Suppose a graph suggests one output value from the left, and a table suggests another value from the right. The representations can be compared even though they display different kinds of information. First identify the target input. Then identify which side each piece of evidence describes. Finally, compare the output values the two sides approach.
Record the input \(a\) at which the limit is being estimated.
Use inputs less than \(a\) for left-hand evidence and inputs greater than \(a\) for right-hand evidence.
Trace each graph branch toward \(a\), or examine how table outputs behave as their inputs get closer to \(a\).
If both sides appear to approach the same output, the representations support that two-sided limit. If they approach different outputs, they do not support a two-sided limit.
Why the Two Sides Must Agree
The reason for comparing both sides follows directly from the meaning of “approaches \(a\).” A two-sided approach allows inputs near \(a\) from either direction. Therefore, if all sufficiently nearby outputs are to be close to one number \(L\), then outputs on the left must be close to \(L\), and outputs on the right must also be close to \(L\).
More formally, a two-sided limit equals \(L\) when, for every positive tolerance \(\varepsilon\), the outputs \(f(x)\) are within \(\varepsilon\) of \(L\) whenever \(x\) is sufficiently close to \(a\), with \(x\ne a\). Restricting those nearby inputs to \(x<a\) gives the left-hand condition; restricting them to \(x>a\) gives the right-hand condition. Thus, a two-sided limit requires both one-sided limits to equal \(L\).
Conversely, if both one-sided limits equal \(L\), then for any \(\varepsilon>0\), there is a left-side distance that makes outputs within \(\varepsilon\) of \(L\), and a right-side distance with the same property. Taking the smaller of those two distances makes the condition hold on both sides at once. That is precisely the two-sided limit condition.
Worked Example: A Graph and Table Support the Same Limit
Worked Example: A Graph and Table Support the Same Limit
A graph shows a curve approaching the height \(5\) as \(x\) approaches \(2\) from either side. A table gives these values near \(x=2\):
| Side | \(x\) | \(f(x)\) | Distance from \(5\) |
|---|---|---|---|
| Left | 1.9 | 4.8 | \(|4.8-5|=0.2\) |
| Left | 1.99 | 4.98 | \(|4.98-5|=0.02\) |
| Right | 2.1 | 5.2 | \(|5.2-5|=0.2\) |
| Right | 2.01 | 5.02 | \(|5.02-5|=0.02\) |
For the left-hand samples, \(1.9<2\) and \(1.99<2\). Their outputs are \(4.8\) and \(4.98\), with distances from \(5\) of \(0.2\) and \(0.02\). The inputs move closer to \(2\), and the outputs move closer to \(5\).
For the right-hand samples, \(2.1>2\) and \(2.01>2\). Their outputs are \(5.2\) and \(5.02\), also with distances from \(5\) of \(0.2\) and \(0.02\). These outputs move closer to \(5\) as the inputs move closer to \(2\).
The graph suggests approach to \(5\) from both sides, and the table’s left and right samples are consistent with that estimate. Check: Both one-sided estimates are \(5\), so they match and support \(\lim_{x\to2}f(x)=5\). The table is evidence from selected inputs, not by itself a proof that the limit exists.
Worked Example: Matching a Left-Hand Graph to Right-Hand Table Values
Worked Example: Matching a Left-Hand Graph to Right-Hand Table Values
Near \(x=1\), a graph shows the branch on the left approaching height \(2\). A table samples the function on the right:
| \(x\) | \(f(x)\) | Distance from \(2\) |
|---|---|---|
| 1.1 | 2.4 | \(|2.4-2|=0.4\) |
| 1.01 | 2.04 | \(|2.04-2|=0.04\) |
| 1.001 | 2.004 | \(|2.004-2|=0.004\) |
Each table input is greater than \(1\), so the data describe approach from the right. The input distances from \(1\) are \(0.1\), \(0.01\), and \(0.001\). The corresponding output distances from \(2\) are \(0.4\), \(0.04\), and \(0.004\), which decrease by a factor of \(10\) at each row.
The table therefore supports a right-hand estimate of \(2\). The graph supplies the other side: its left branch approaches \(2\). The representations match at the output value \(2\).
Check: The graph’s left-hand estimate and the table’s right-hand estimate agree. Together they support \(\lim_{x\to1}f(x)=2\); the table values are not exactly \(2\), but their distances from \(2\) shrink as the inputs approach \(1\).
Worked Example: The Representations Do Not Match
Worked Example: The Representations Do Not Match
At \(x=0\), a graph shows its left branch approaching height \(4\). A table gives samples from the right:
| \(x\) | \(f(x)\) | Distance from \(4\) | Distance from \(3\) |
|---|---|---|---|
| 0.1 | 3.2 | \(|3.2-4|=0.8\) | \(|3.2-3|=0.2\) |
| 0.01 | 3.02 | \(|3.02-4|=0.98\) | \(|3.02-3|=0.02\) |
| 0.001 | 3.002 | \(|3.002-4|=0.998\) | \(|3.002-3|=0.002\) |
All the table inputs are greater than \(0\), and their distances from \(0\) shrink. The outputs become closer to \(3\): their distances from \(3\) are \(0.2\), \(0.02\), and \(0.002\). Their distances from \(4\), by contrast, do not shrink toward zero. The table supports a right-hand estimate of \(3\).
The graph’s left-hand estimate is \(4\), while the table’s right-hand estimate is \(3\). Since \(4\ne3\), the two sides do not approach the same value.
Check: The one-sided estimates are unequal, so the representations do not support a two-sided limit at \(x=0\). A two-sided limit cannot equal both \(4\) and \(3\).
A Common Mistake: Expecting Table Entries to Equal the Limit
A limit describes the value that outputs approach; it does not require the sampled outputs to equal that value. In the second example, the table entries \(2.4\), \(2.04\), and \(2.004\) are not equal to \(2\). That does not contradict the estimate, because their distances from \(2\) shrink as the inputs approach the target.
This compares one particular table output with the graph’s estimated approach value, rather than comparing the trends. The corrected comparison is: the table’s outputs have distances \(0.4\), \(0.04\), and \(0.004\) from \(2\), so the right-hand values move toward \(2\); the graph’s left branch also approaches \(2\). The estimates agree.
Reading the Evidence Carefully
A graph and a finite table provide estimates, not a guarantee about every input arbitrarily close to the target. Graphs have limited visual precision, and tables display only selected inputs. Use them to decide what the evidence supports, and be explicit about which side each representation covers.
- A left-side table must use inputs less than the target; a right-side table must use inputs greater than it.
- Compare the output values being approached, not whether the table entries are identical.
- Check that the graph trace is followed toward the target on the correct side.
- Matching estimates support a common two-sided limit; different one-sided estimates do not.
Check Your Understanding
For each question, identify the left- and right-hand evidence before deciding what it supports.
- A graph approaches height \(6\) from the left of \(x=3\). A right-hand table has outputs \(6.3\), \(6.03\), and \(6.003\) at inputs \(3.1\), \(3.01\), and \(3.001\). What two-sided limit do these representations support?
- A left-hand table’s outputs approach \(-1\), while the graph’s right branch approaches \(2\). Do these estimates support a two-sided limit? Explain.
- Why is an output value of \(4.98\) in a table not necessarily evidence against a limit of \(5\)?
- What must be true of the left- and right-hand limits for the two-sided limit at \(x=a\) to equal \(L\)?
