What a Break Does—and Does Not—Tell You
A graph can have a hole, a jump, or branches that do not meet at the target. Such a break shows that something happens at that input, but it does not, by itself, tell you whether the limit exists. To read a limit, track the graph’s height as the input moves toward the target from nearby values on each side.
The graph’s point exactly at the target is a separate piece of information. A filled point tells you the function value there, if the graph represents a function defined at that input. An open circle indicates that the point is not included at that location. Neither marker alone determines the height the nearby branches approach.
Read Each Side Toward the Target
The left-hand limit, written \(\lim_{x\to a^-}f(x)\), is the height the graph approaches as \(x\) approaches \(a\) through values less than \(a\). The right-hand limit, \(\lim_{x\to a^+}f(x)\), is the height it approaches through values greater than \(a\). The superscript signs tell you which side of the target to trace; they do not say whether the graph is above or below the horizontal axis.
The two-sided limit, \(\lim_{x\to a}f(x)\), exists when the left-hand and right-hand limits both exist and equal the same finite number. If they approach different heights, there is no two-sided limit, even if the graph has a filled point at \(x=a\).
When estimating from a graph, read the vertical coordinate of the branch near the target. Do not read the horizontal coordinate of an open circle as the limit; the limit is a height, or \(y\)-value. If the graph has a grid, use its scale carefully. A branch approaching a height between grid lines may support an estimate rather than an exact value.
A Reliable Graph-Reading Process
Use this sequence whenever the graph is broken at the target. It helps prevent a conspicuous point or gap from distracting you from the nearby behavior.
Mark \(x=a\) on the horizontal axis. Keep the task focused on what happens as inputs approach this value.
Follow the branch for \(x<a\) toward \(a\). Estimate the height it approaches, and record that as the left-hand limit.
Follow the branch for \(x>a\) toward \(a\). Estimate its approaching height, and record that as the right-hand limit.
If both sides approach the same finite height, that height is the two-sided limit. If the heights differ, the two-sided limit does not exist.
If asked for \(f(a)\), look for a filled point at \(x=a\). Do not substitute that point’s height for an approaching height.
Why the Two Sides Must Agree
The comparison rule follows from what it means for a graph to approach one height. Suppose the left and right branches both approach \(L\). For any chosen small vertical tolerance \(\varepsilon>0\), each side gets within \(\varepsilon\) of \(L\) once its input is sufficiently close to \(a\). Choose a distance \(\delta_-\) that works on the left and a distance \(\delta_+\) that works on the right. Taking the smaller of those distances ensures both sides are within the tolerance whenever \(0<|x-a|<\delta\), where \(\delta=\min(\delta_-,\delta_+)\). Thus the graph approaches \(L\) from both directions.
Conversely, if the graph approaches a single height \(L\) from both sides, then restricting attention to inputs less than \(a\) gives the left-hand limit \(L\), and restricting attention to inputs greater than \(a\) gives the right-hand limit \(L\). Therefore a two-sided limit exists exactly when the two one-sided limits exist and match. The value at \(x=a\) is not part of this comparison, because approaching \(a\) uses nearby inputs, not the input \(a\) itself.
Worked Examples
Worked Example: A Hole with a Different Function Value
Suppose a graph has a straight branch on each side of \(x=2\). Both branches lie along the same line and approach the open circle \((2,3)\). There is also a filled point at \((2,-1)\). Estimate the one-sided limits, the two-sided limit, and the function value at \(2\).
Step 1: Trace from the left. The branch approaches height \(3\). For instance, the graph has height \(2.9\) at \(x=1.9\) and height \(2.99\) at \(x=1.99\). These heights move toward \(3\), so \(\lim_{x\to 2^-}f(x)=3\).
Step 2: Trace from the right. The branch approaches the same open circle. The graph has height \(3.01\) at \(x=2.01\) and height \(3.1\) at \(x=2.1\). As the inputs move closer to \(2\), the heights move toward \(3\), so \(\lim_{x\to 2^+}f(x)=3\).
Step 3: Compare the sides. Both one-sided limits equal \(3\), so \(\lim_{x\to 2}f(x)=3\).
Step 4: Read the point. The filled point at \(x=2\) has height \(-1\), so \(f(2)=-1\). The approaching height \(3\) and the function value \(-1\) answer different questions.
Check: The left and right branch readings agree at \(3\), while the filled point is at \(-1\). This matches the conclusion that the limit is \(3\) and the function value is \(-1\).
Worked Example: Separated Branches with a Shared Approaching Height
Consider a graph near \(x=0\). Its left branch ends in an open circle at \((0,-2)\), and its right branch begins at a separate open circle at \((0,-2)\). The two branches have different shapes. A filled point is also plotted at \((0,1)\). Estimate the limits and the function value.
Step 1: Read the left branch. On the left, the graph is at height \(-1.8\) when \(x=-0.1\), and at height \(-1.98\) when \(x=-0.01\). The latter input is closer to \(0\), and its height is closer to \(-2\). The branch approaches \(-2\), so \(\lim_{x\to 0^-}f(x)=-2\).
Step 2: Read the right branch. On the right, the graph is at height \(-2.3\) when \(x=0.1\), and at height \(-2.03\) when \(x=0.01\). As the input moves closer to \(0\), the height moves closer to \(-2\). Thus \(\lim_{x\to 0^+}f(x)=-2\).
Step 3: Compare the sides. The left-hand and right-hand limits are both \(-2\). Therefore \(\lim_{x\to 0}f(x)=-2\), even though the branches are not joined by a filled point at the target.
Step 4: Read the filled point. The filled point is at \((0,1)\), so \(f(0)=1\). It does not change either approaching height.
Check: Left-hand height \(-2\) equals right-hand height \(-2\), so the two-sided limit is \(-2\). The separate filled point gives \(f(0)=1\), not the limit.
Worked Example: A Jump with Different One-Sided Limits
Near \(x=1\), suppose the left branch approaches height \(4\), while the right branch approaches height \(1\). A filled point is plotted at \((1,4)\). Determine whether the two-sided limit exists.
Step 1: Read from the left. The left branch’s heights approach \(4\). For example, its height is \(4.1\) at \(x=0.9\) and \(4.01\) at \(x=0.99\). Thus \(\lim_{x\to 1^-}f(x)=4\).
Step 2: Read from the right. The right branch’s heights approach \(1\). For example, its height is \(1.2\) at \(x=1.1\) and \(1.02\) at \(x=1.01\). Thus \(\lim_{x\to 1^+}f(x)=1\).
Step 3: Compare. Since \(4\ne 1\), the one-sided limits disagree. Therefore \(\lim_{x\to 1}f(x)\) does not exist.
Step 4: Read the point separately. The filled point gives \(f(1)=4\). Its height agrees with the left-hand limit, but it cannot make the right-hand limit equal \(4\).
Check: The two traces give \(4\) and \(1\), which are unequal. The graph’s filled point at height \(4\) does not alter that comparison, so the two-sided limit does not exist.
A Common Mistake: Treating Every Break as a Failed Limit
A tempting but incorrect conclusion is: “There is an open circle at the target, so the limit does not exist.” For example, in the first worked example, that reasoning would claim that \(\lim_{x\to 2}f(x)\) does not exist because the graph has an open circle at \((2,3)\). This is wrong: tracing both branches shows that each approaches \(3\), so the limit is \(3\).
The opposite mistake is also possible: “There is a filled point at the target, so the limit equals its height.” In the jump example, that would give a limit of \(4\) from the filled point, even though the right branch approaches \(1\). The correction in both cases is the same: trace each nearby branch toward the target, record the two approaching heights, and compare them. Use the filled point only when the question asks for the function value.
What Graph Evidence Can Support
A graph usually supports an estimate, not unlimited precision. If the branches appear to approach \(2.5\), report that as an estimated height unless the graph provides enough information to identify an exact value. Do not claim that a limit is exactly a particular number merely because one nearby point has that height; the trend toward the target is the evidence that matters.
Also check that there is a branch on each side arbitrarily close to the target before making a two-sided comparison. If the displayed graph is cut off, or one side has no visible nearby branch, the picture may not provide enough information to judge that side. A missing piece of drawing is not, by itself, proof of any particular limit behavior.
These habits make broken graphs easier to interpret: a hole can coexist with a limit, separated branches can approach the same height, and a jump can prevent a two-sided limit. The decision comes from the two approaching heights—not from the appearance of a break alone.
Check Your Understanding
For each question, trace the branches toward the target before using any point marked exactly at that input.
- A graph approaches height \(5\) from the left and height \(5\) from the right at \(x=3\), but has an open circle there. What is the two-sided limit?
- At \(x=-1\), the left branch approaches \(2\) and the right branch approaches \(-2\). Does the two-sided limit exist? Explain using the one-sided limits.
- A filled point at \(x=4\) has height \(7\), while both branches approach height \(1\). State the limit and the function value.
- Why does an open circle at the target not, by itself, show that the limit fails to exist?
- What should you conclude if the graph shows a nearby branch on only one side of the target and provides no information about the other side?
