Reading a Limit From a Graph
A graph can give a quick visual estimate of a limit. The key is to look at what the graph does near the input of interest, not simply at the point directly above that input. A curve may approach a height even if the function is not defined at the input, or even if its value there is somewhere else.
In this workshop, we will read a graph by following its trace from inputs on either side of a target \(x\)-value. Each estimate should be accompanied by the evidence: which branches approach which height, and whether they agree.
This graphical description is an estimate, not by itself a proof. A drawing has finite resolution, and a narrow feature or complicated behavior may not be visible. The graph is evidence for a proposed limit; a proof requires mathematical reasoning about the function.
Open Circles, Filled Points, and Nearby Traces
An open circle marks a point excluded from the plotted graph at that location. It often indicates a height the curve approaches while the function is not assigned that value at the input. A filled point marks a point included in the graph, so if the filled point is at \((a,c)\), then \(f(a)=c\).
Neither marker alone determines the limit. The trace on either side of \(x=a\) is the evidence about nearby behavior. A filled point can be far from the height approached by the curve. An open circle can mark the limiting height, but only if the nearby trace actually approaches it.
Read Each Side of the Graph
The graph may have a different trace to the left of \(a\) and to the right of \(a\). The left-hand limit, written \(\lim_{x\to a^-}f(x)\), describes the height approached as \(x\) approaches \(a\) through values less than \(a\). The right-hand limit, written \(\lim_{x\to a^+}f(x)\), describes the height approached through values greater than \(a\).
A two-sided limit \(\lim_{x\to a}f(x)\) exists only when both sides approach the same finite height. If the left trace approaches one height and the right trace approaches another, there is no single two-sided limit. The value \(f(a)\), if defined, does not make unequal side estimates agree.
Why the Two Sides Must Agree
The requirement that both sides approach \(L\) follows from the epsilon-delta definition of a limit. In that definition, for every \(\varepsilon>0\), there must be a \(\delta>0\) such that whenever \(0<|x-a|<\delta\), we have \(|f(x)-L|<\varepsilon\). The condition \(0<|x-a|\) includes inputs on both sides of \(a\), while excluding \(a\) itself.
Here is the full reasoning behind the graphical criterion. If the two-sided limit is \(L\), then the definition applies to every nearby \(x\) on the left and every nearby \(x\) on the right. Thus each side must approach \(L\). Conversely, suppose both side limits are \(L\). For a chosen \(\varepsilon>0\), the left-side limit supplies a distance \(\delta_1>0\) that keeps the graph within \(\varepsilon\) of \(L\) for inputs just left of \(a\). The right-side limit supplies a distance \(\delta_2>0\) with the same property just right of \(a\). Choose the smaller distance. Then every \(x\) within that distance of \(a\), other than \(a\), is on one of those two sides, and its graph height is within \(\varepsilon\) of \(L\). That is exactly the two-sided limit condition.
A Practical Graph-Reading Method
Use the following steps when a graph asks for a limit. Take care to describe the evidence rather than reporting only a number.
Find \(x=a\) on the horizontal axis and focus on a small region around it.
Follow graph points with \(x<a\) toward \(a\), and estimate the height they approach.
Follow graph points with \(x>a\) toward \(a\), and estimate that height separately.
If both traces approach the same height, report it as the two-sided limit estimate. If they differ, do not report a two-sided limit.
Record \(f(a)\) only if asked for the function value; do not use it as a substitute for the trace evidence.
Worked Examples
Worked Example: A Hole With a Different Function Value
Graph description: For inputs just less than \(2\), the curve rises toward the open circle \((2,3)\). For inputs just greater than \(2\), it also approaches that same open circle. There is a separate filled point at \((2,1)\).
Step 1: Read the left trace. As \(x\) approaches \(2\) from below, the curve’s height approaches \(3\). Thus the left-side estimate is \(3\).
Step 2: Read the right trace. As \(x\) approaches \(2\) from above, the curve’s height also approaches \(3\). Thus the right-side estimate is \(3\).
Step 3: Compare the sides. Both traces approach the same height, so the graph supports the estimate \(\lim_{x\to 2}f(x)=3\).
Step 4: Check the point value. The filled point is \((2,1)\), so \(f(2)=1\). This does not change the limit estimate. The check is consistent: the two branches approach height \(3\), while the graph assigns height \(1\) at the target input.
Worked Example: Different Heights on the Two Sides
Graph description: To the left of \(x=-1\), the curve approaches height \(2\), marked by an open circle at \((-1,2)\). To the right, a separate branch approaches height \(4\), marked by an open circle at \((-1,4)\). A filled point is plotted at \((-1,2)\).
Step 1: Read the left trace. Following the graph from \(x<-1\) toward \(-1\), the heights approach \(2\). Therefore \(\lim_{x\to -1^-}f(x)\approx 2\).
Step 2: Read the right trace. Following the branch from \(x>-1\) toward \(-1\), the heights approach \(4\). Therefore \(\lim_{x\to -1^+}f(x)\approx 4\).
Step 3: Compare the side estimates. Since \(2\ne4\), there is no single height approached from both sides. The graph does not support a two-sided limit at \(-1\).
Step 4: Check the filled point. The filled point gives \(f(-1)=2\), agreeing with the left-side height in this drawing. It does not alter the right-side estimate of \(4\), so it cannot create a two-sided limit.
Worked Example: Matching Traces With a Separate Filled Point
Graph description: At \(x=0\), an open circle is located at height \(-2\). The curve to the left and the curve to the right both move toward that open circle as their inputs approach \(0\). A filled point is located at \((0,3)\).
Step 1: Estimate from the left. The left branch approaches height \(-2\), so \(\lim_{x\to 0^-}f(x)\approx -2\).
Step 2: Estimate from the right. The right branch also approaches height \(-2\), so \(\lim_{x\to 0^+}f(x)\approx -2\).
Step 3: Determine the two-sided estimate. The side estimates match, so \(\lim_{x\to 0}f(x)\approx -2\).
Step 4: Check the plotted value. The filled point gives \(f(0)=3\), which differs from \(-2\). The estimate still follows from the two matching nearby traces, not from the filled point.
A Common Mistake: Reading the Filled Point as the Limit
Suppose a graph has a filled point at \((2,1)\), while the curve on both sides approaches the open circle \((2,3)\). A common incorrect calculation is:
The implication is invalid. The first statement reports the function’s value at exactly \(x=2\); it says nothing by itself about nearby inputs. The correction is to inspect the traces: the left branch approaches \(3\), and the right branch approaches \(3\). Therefore the graph suggests \(\lim_{x\to 2}f(x)=3\), while \(f(2)=1\).
A second visual caution is that a graph can suggest a height without establishing it exactly. If the vertical scale is coarse, heights such as \(2.9\) and \(3.1\) might be difficult to distinguish. State an estimate supported by the plotted scale, and avoid claiming that a picture alone proves exact equality.
What a Graph Can and Cannot Tell You
A graph is especially useful for forming a limit estimate and identifying the relevant evidence. It can show that both visible branches appear to approach one height, that the sides appear to approach different heights, or that a point value differs from the nearby behavior.
But the conclusion must be no stronger than the graph warrants. A drawn curve represents a function at a limited level of visual precision. For an exact claim, use a formula, a precise description of the graph, or a proof based on the limit definition. The discipline of separating observation from proof is important: first state what the picture suggests, then identify what would be needed to establish the claim mathematically.
Check Your Understanding
For each question, state which graph feature supplies the evidence and distinguish the limit from the function value when relevant.
- A curve on both sides of \(x=4\) approaches an open circle at \((4,5)\), and a filled point is at \((4,0)\). What limit estimate is supported, and what is \(f(4)\)?
- As \(x\) approaches \(1\) from the left, a branch approaches height \(-3\); from the right, it approaches height \(2\). What can you conclude about the two-sided limit?
- What does a filled point at \((a,c)\) tell you, and what does it not tell you by itself?
- Why must the left- and right-side traces approach the same height for a two-sided limit to exist?
- Why should a graph-based estimate not automatically be described as a proof?
