Reading a Limit From a Graph
A graph gives a visual record of how the outputs of a function change as the input changes. To estimate a limit from a graph, choose the target input, then trace the curve toward that input from nearby values. The height the curve approaches is the estimate of the limit.
The key is to follow the curve near the target, rather than to focus only on what is drawn exactly at the target. A point at the target records the function’s value there, if the function has one. Nearby parts of the curve show the behavior relevant to the limit.
This is an estimate based on the graph, not necessarily an exact calculation. A drawing can suggest a value clearly, but a graph alone may not establish an exact value or prove that a limit exists.
Three Pieces of Information to Keep Separate
When reading a graph near \(x=a\), distinguish these features:
- The target input: the horizontal coordinate \(a\) toward which the input moves.
- The nearby curve: the graph’s path as \(x\) gets closer to \(a\), without requiring \(x\) to equal \(a\).
- The point at the target: a filled point at \(x=a\) indicates the function’s value there. If there is no point, the function may be undefined at that input.
An open circle marks a location that the curve approaches but that is not included as a point of the graph at that coordinate. A filled point marks a point that is included. These markers can occur at different heights for the same input. That difference does not, by itself, tell us the limit; trace the nearby curve to estimate the limiting height.
Worked Example: A Curve Passing Through the Target
Suppose a graph shows a straight curve near \(x=2\). On the left, it passes through approximately \((1.8,2.6)\) and \((1.9,2.8)\); on the right, it passes through approximately \((2.1,3.2)\) and \((2.2,3.4)\). The curve passes through \((2,3)\).
Step 1: Locate the target input. The limit asks about inputs approaching \(2\), so focus on the curve near the vertical line \(x=2\).
Step 2: Trace from the left. As the displayed inputs increase from \(1.8\) to \(1.9\) and toward \(2\), the heights increase from about \(2.6\) to \(2.8\) and toward \(3\).
Step 3: Trace from the right. As the inputs decrease from \(2.2\) to \(2.1\) and toward \(2\), the heights decrease from about \(3.4\) to \(3.2\) and toward \(3\).
Estimate: Both traces approach height \(3\), so the graph supports the estimate \(\lim_{x\to 2}f(x)\approx 3\). The curve also includes the point \((2,3)\), so this graph indicates \(f(2)=3\). The limit estimate and the function value happen to agree.
Check: The left-side heights and right-side heights both head toward \(3\). The estimate is consistent with the point on the curve at the target.
Why Tracing the Curve Works
The meaning of a limit concerns outputs at inputs close to \(a\), not just the output at \(a\). The graph depicts these input-output pairs: a point on the curve has horizontal coordinate \(x\) and vertical coordinate \(f(x)\). Moving along the curve toward the vertical line \(x=a\) therefore shows how the outputs behave as the inputs approach \(a\).
In particular, if the curve’s nearby heights settle toward a common level \(L\) from both sides, then \(L\) is the graphical estimate of the limit. The estimate concerns the behavior around the target, so a point plotted exactly at \(x=a\) can have a different height without changing what the nearby curve approaches.
Worked Example: The Function Value Differs From the Limit
Suppose the graph near \(x=1\) follows the straight line through \((0.8,1.6)\), \((0.9,1.8)\), \((1.1,2.2)\), and \((1.2,2.4)\). The line has an open circle at \((1,2)\). A separate filled point is plotted at \((1,-1)\).
Step 1: Read the nearby curve on the left. At \(x=0.8\), the height is \(1.6\); at \(x=0.9\), it is \(1.8\). Following the line toward \(x=1\), these heights move toward \(2\).
Step 2: Read the nearby curve on the right. At \(x=1.2\), the height is \(2.4\); at \(x=1.1\), it is \(2.2\). Following the line back toward \(x=1\), these heights also move toward \(2\).
Step 3: Read the point at the target separately. The open circle at \((1,2)\) is approached by the line but is not included as a graph point. The filled point at \((1,-1)\) indicates that \(f(1)=-1\).
Estimate: The curve approaches height \(2\) from both sides, so \(\lim_{x\to 1}f(x)\approx 2\), while the graph gives \(f(1)=-1\).
Check: The limit estimate matches the height of the open circle approached by the curve, not the height of the filled point. The two values answer different questions, so there is no contradiction.
Estimating a Height That Is Not Marked Exactly
Sometimes a graph does not place a labelled point precisely at the height the curve approaches. In that case, use the nearby grid marks as guides and state the result as an estimate. For example, the curve may appear to approach a height between \(1\) and \(2\). Do not report a more precise number than the graph supports.
It is helpful to make a separate estimate from each side before deciding on a two-sided estimate. This guards against being distracted by a point at the target and gives a simple consistency check: the two traces should head toward the same height.
Worked Example: Estimating Between Grid Lines
Suppose a graph has horizontal grid lines at integer heights. Near \(x=-1\), a smooth curve approaches a level that appears halfway between the lines \(y=1\) and \(y=2\). On the left of \(x=-1\), it rises toward that level; on the right, it falls toward the same level. There is an open circle at the apparent meeting point, and no filled point at \(x=-1\).
Step 1: Locate the target. The input is approaching \(-1\), so follow the curve close to the vertical line \(x=-1\).
Step 2: Estimate the left-hand trace. The left branch moves toward a height halfway between \(1\) and \(2\), which is approximately \(1.5\).
Step 3: Estimate the right-hand trace. The right branch appears to approach that same halfway height, also approximately \(1.5\).
Step 4: Interpret the marker. The open circle shows the point being approached, but it does not define a function value there. With no filled point at \(x=-1\), this graph does not show a value \(f(-1)\).
Estimate: The graph suggests \(\lim_{x\to -1}f(x)\approx 1.5\).
Check: Both branches approach the same apparent height, so the two-sided estimate is consistent. Because the height is read between grid lines, \(1.5\) is an estimate from the drawing rather than an exact value established by it.
A Practical Graph-Reading Process
Use the following sequence whenever you estimate a limit from a graph:
Locate \(a\) on the horizontal axis and identify the nearby part of the graph.
Follow points with inputs less than \(a\) toward the target and estimate their approaching height.
Follow points with inputs greater than \(a\) toward the target and estimate their approaching height.
If both traces approach the same height, use it as the estimate for the two-sided limit.
Record a filled point as the function value, if present, but do not substitute it for the nearby curve’s behavior.
Common Mistake: Using the Filled Point as the Limit
A frequent error is to read the height of the filled point at \(x=a\) and report it as the limit, without tracing the nearby curve. For the graph in the previous worked example, that mistake would produce:
This conclusion confuses the function value with the limiting height. The filled point at \((1,-1)\) tells us that \(f(1)=-1\), but the nearby curve approaches height \(2\). The corrected reading is:
Correction: First trace the curve toward the target from nearby inputs. Then inspect the point at the target as a separate piece of information. An open circle, a filled point, or the absence of a point does not replace this tracing step.
What a Graph Can and Cannot Establish
A graph is useful because it makes local behavior visible. It can help you identify a plausible limiting height, notice whether traces from both sides agree, and avoid confusing that height with the function value at the target.
However, a visual estimate should be described at an appropriate level of precision. If the curve seems to approach \(2\), say that the graph suggests a limit of \(2\); do not claim that the drawing alone proves an exact mathematical value. A graph may be approximate, and its appearance only supplies evidence about the behavior it depicts. For now, the essential skill is to read that evidence carefully: follow the curve near the target, compare its approaching heights, and keep the target point distinct.
Check Your Understanding
Use the graph descriptions to answer each question. State an estimate where appropriate and explain which part of the graph supports it.
- A curve approaches height \(4\) from both sides of \(x=3\), and passes through \((3,4)\). What limit does the graph suggest, and what is the function value?
- A curve approaches an open circle at \((2,5)\), while a filled point is at \((2,1)\). What does the graph suggest for the limit and for the function value?
- Why should you trace the curve near a target input instead of reading only the filled point at that input?
- A curve’s left and right portions both appear to approach a height halfway between \(y=0\) and \(y=1\). What is a reasonable graphical estimate, and why should it be described as an estimate?
- What does an open circle tell you, and what does it not tell you by itself?
