Reading Behavior Near an Input
A graph or table can show how a function behaves near an input, even when the function’s value at that input is unusual or is not defined. The key question is not necessarily “What is the function’s value here?” Instead, it is “What values does the function approach as the input comes close to this point from one specified side?”
A left-hand limit uses inputs smaller than the point of interest. A right-hand limit uses inputs larger than it. Reading the two sides separately helps make sense of jumps, holes, and other situations in which behavior on the two sides may differ.
More precisely, for the left-hand limit, for every \(\varepsilon>0\) there must be a \(\delta>0\) such that whenever \(x\) is in the domain of \(f\) and \(a-\delta<x<a\), we have \(|f(x)-L|<\varepsilon\). For the right-hand limit, the corresponding condition is \(a<x<a+\delta\). The inequalities \(x<a\) and \(x>a\) select the side; the input \(x=a\) is not included.
Read the Side Before Reading the Height
On a graph, locate the input \(a\) on the horizontal axis. To estimate a left-hand limit, follow the graph as its input approaches \(a\) from smaller values. To estimate a right-hand limit, follow it from larger values. Track the height the graph approaches, not merely the point plotted at \(x=a\).
An open circle often marks a point the graph approaches but does not include. A filled point marks a function value. These marks are useful clues, but the surrounding graph determines the limit. In particular, a filled point at \(x=a\) need not lie at the height approached by the graph from either side.
Worked Example: Reading a Limit from a Graph
Worked Example: Reading a Limit from a Graph
Suppose a graph has a curve approaching the open circle \((2,5)\) from the left. It also has a curve approaching that same open circle from the right. A filled point is plotted at \((2,1)\). Estimate both one-sided limits and state the function value.
From the left, the curve’s height approaches \(5\) as the input approaches \(2\) through values less than \(2\). Therefore, \(\lim_{x\to 2^-}f(x)=5\).
From the right, the curve’s height also approaches \(5\) as the input approaches \(2\) through values greater than \(2\). Therefore, \(\lim_{x\to 2^+}f(x)=5\).
The filled point is at \((2,1)\), so \(f(2)=1\). This does not change either limit, because neither one uses the input \(x=2\).
Check: The left- and right-side graph traces both approach height \(5\), while the filled point gives height \(1\) exactly at input \(2\). Thus the two limit estimates agree with the graph, and the function value is separately \(1\).
Estimating Limits from Tables
A numerical table gives sample inputs and their corresponding function values. For a left-hand estimate at \(a\), inspect inputs below \(a\) that move closer to it. For a right-hand estimate, inspect inputs above \(a\) that move closer. Then check whether the corresponding function values appear to settle near one number.
For example, values approaching \(4\) from below might include \(3.9\), \(3.99\), and \(3.999\). Values approaching from above might include \(4.1\), \(4.01\), and \(4.001\). The direction of the inputs matters: a collection of values on only one side supports an estimate for that side, not automatically for the other.
Worked Example: Estimating from a Table
The table gives values for a function near \(x=3\). Estimate the left- and right-hand limits.
| Input \(x\) | Function value \(f(x)\) |
|---|---|
| 2.9 | 6.8 |
| 2.99 | 6.98 |
| 2.999 | 6.998 |
| 3.001 | 7.002 |
| 3.01 | 7.02 |
| 3.1 | 7.2 |
The first three inputs are less than \(3\), and they approach \(3\). Their function values are \(6.8\), \(6.98\), and \(6.998\), which approach \(7\). The table suggests \(\lim_{x\to 3^-}f(x)=7\).
The last three inputs are greater than \(3\), and they approach \(3\). Their function values are \(7.2\), \(7.02\), and \(7.002\), which also approach \(7\). The table suggests \(\lim_{x\to 3^+}f(x)=7\).
Check: The left-side values are below \(7\) and get closer to it; the right-side values are above \(7\) and get closer to it. Both sets support the estimate \(7\). A finite table provides evidence, not a proof that all sufficiently nearby values behave this way.
Compare the Two Sides Separately
Do not assume that the left-hand and right-hand limits agree. Read each side independently, then compare the estimates. If they approach different finite values, there is no single finite value approached from both sides. This comparison is useful even when the task asks only for one one-sided limit: it prevents information from one side being mistaken for information about the other.
Worked Example: Different Estimates from the Two Sides
A graph approaches height \(2\) as \(x\) approaches \(1\) from the left, and approaches height \(5\) as \(x\) approaches \(1\) from the right. A filled point at \(x=1\) has height \(4\). State the one-sided limits and explain what the picture shows.
The left trace approaches \(2\), so \(\lim_{x\to 1^-}f(x)=2\).
The right trace approaches \(5\), so \(\lim_{x\to 1^+}f(x)=5\).
The filled point gives \(f(1)=4\). It does not make either side approach \(4\): the left and right traces still approach heights \(2\) and \(5\), respectively.
Check: Approaching from \(x<1\) gives the height \(2\); approaching from \(x>1\) gives the height \(5\). Since those estimates differ, they cannot be replaced by a single shared value. The function value at \(1\) is \(4\), a third, separate piece of information.
Why the Function Value Does Not Set the Limit
The one-sided definition explains why the point value is separate. For the left-hand limit, the condition is tested only for \(x<a\); for the right-hand limit, it is tested only for \(x>a\). In both cases, \(x=a\) is excluded. Changing \(f(a)\) while leaving all nearby function values unchanged therefore leaves both one-sided limits unchanged.
For example, in the first graph example the filled point could be moved from height \(1\) to height \(8\), or removed, without changing the traces approaching height \(5\). The estimates of the limits would still be \(5\) from both sides. This is why a graph should be read by following its nearby curves, rather than by treating the filled point as the answer.
How Reliable Is an Estimate?
A graph gives a visual estimate, and a table gives a finite selection of numerical evidence. Both can be misleading if viewed carelessly. A graph may have limited resolution; a table may omit values that behave differently even closer to \(a\). So report what the evidence supports, and distinguish an estimate from a proven limit.
When reading a table, ask whether the inputs genuinely move toward \(a\) from the required side. Then check whether the output values move toward a stable candidate. If the output values fluctuate or trend away, do not force them into a single estimate. When reading a graph, trace the branch near the input of interest and look at its height as it approaches that input.
Mark \(a\) on the horizontal axis or identify it in the table.
Use inputs less than \(a\) for a left-hand limit, or greater than \(a\) for a right-hand limit.
Follow the nearby graph heights or table outputs as the inputs get closer to \(a\).
Report the height approached on that side. If \(f(a)\) is shown, record it as a separate value.
Common Mistake: Using the Filled Point as the Limit
Suppose a graph approaches height \(6\) from the left and has a filled point at \((4,2)\). A mistaken calculation is: “The point at \(x=4\) has height \(2\), so \(\lim_{x\to4^-}f(x)=2\).” This confuses the value at the input with the nearby behavior.
The correction is to follow the graph from inputs less than \(4\). If its heights approach \(6\), then \(\lim_{x\to4^-}f(x)=6\), while the filled point says \(f(4)=2\). The point value belongs in the answer only when the question asks for \(f(4)\).
Summary
A one-sided limit is an estimate of the function value approached as inputs move toward a specified point from just one direction. Read the left and right sides separately, using nearby graph traces or table entries. Open circles and filled points can help interpret a graph, but the branch near the input determines the limit. The value at the input may differ from either one-sided limit, and a graph or finite table provides evidence rather than a complete proof.
Check Your Understanding
For each question, explain which side or information is relevant before giving an estimate.
- A graph approaches height \(3\) from the left of \(x=2\), and a filled point is at \((2,7)\). What are the left-hand limit estimate and \(f(2)\)?
- A table lists inputs \(0.9\), \(0.99\), and \(0.999\) near \(1\). Which one-sided limit can these entries help estimate, and why?
- A graph approaches height \(4\) from the right of \(x=5\). What is the right-hand limit estimate?
- A graph approaches height \(1\) from the left of \(x=3\) and height \(6\) from the right. What does comparing these estimates tell you?
- Why can changing only the value \(f(a)\) leave both one-sided limits unchanged?
