Why Compare the Two Sides?
A two-sided limit describes the value a function approaches as the input gets close to a target from either direction. The function may approach the same height from both sides, or it may approach different heights. Comparing those two behaviors is the key test for deciding whether a finite two-sided limit exists.
For example, as \(x\) approaches \(c\), the graph might approach height \(4\) from the left and height \(4\) from the right, even if the function is not defined at \(c\). In that case, the two-sided limit is \(4\). But if the left side approaches \(4\) and the right side approaches \(7\), there is no single value approached from both directions, so the finite two-sided limit does not exist.
What the Two-Sided Limit Means
The notation \(\lim_{x\to c} f(x)=L\) says that the values \(f(x)\) approach \(L\) as \(x\) gets close to \(c\), with \(x\) allowed to approach from either side. The input \(x=c\) itself is excluded from this approaching behavior. Therefore, the value \(f(c)\), if it is defined, need not equal \(L\).
The left-hand limit, written \(\lim_{x\to c^-}f(x)\), describes the behavior for inputs less than \(c\) and close to \(c\). The right-hand limit, written \(\lim_{x\to c^+}f(x)\), describes the behavior for inputs greater than \(c\) and close to \(c\). Each one-sided limit must approach a single finite number for it to serve as a finite value in the criterion.
This definition includes inputs on both sides of \(c\). A one-sided limit uses the same closeness idea but restricts \(x\) to one side: \(c-\delta<x<c\) for the left-hand limit, or \(c<x<c+\delta\) for the right-hand limit.
The Equality Test
To decide whether the two-sided limit exists, determine what value the function approaches from each side. Then compare the results. If both one-sided limits exist as finite values and match, that shared value is the two-sided limit. If the values differ, there is no finite two-sided limit.
The criterion is an “if and only if” statement. Each direction matters: a two-sided limit forces both one-sided limits to equal it, and matching one-sided limits combine to give the two-sided limit.
Why the Criterion Works
Here is a justification using the definition. Suppose first that the two-sided limit is \(L\). For every sufficiently close \(x\ne c\), the values \(f(x)\) are close to \(L\). This applies in particular to inputs just below \(c\), and also to inputs just above \(c\). Therefore each one-sided limit equals \(L\).
Conversely, suppose both one-sided limits equal the same finite number \(L\). Given any \(\varepsilon>0\), the left-hand limit supplies a positive distance \(\delta_{\text{left}}\) such that inputs on the left within that distance satisfy \(|f(x)-L|<\varepsilon\). The right-hand limit supplies a positive distance \(\delta_{\text{right}}\) with the same property on the right. Choose the smaller distance. Then every input on either side within that distance satisfies the required inequality.
Whenever \(0<|x-c|<\delta\), the input is either left of \(c\) or right of \(c\). In the first case the left-hand condition applies; in the second case the right-hand condition applies. Thus \(|f(x)-L|<\varepsilon\) for all such \(x\), which is exactly the definition of the two-sided limit.
Reading the Test from a Graph
On a graph, trace each branch toward \(x=c\), without requiring the graph to include the point \(x=c\). Ask what height the branch approaches from the left, then ask what height it approaches from the right. The height of a filled point at \(c\) gives \(f(c)\), not necessarily the limit.
Worked Example: Matching Graph Heights Despite a Different Function Value
Suppose a graph has a branch on the left that approaches height \(2\) as \(x\to 3^-\), and a branch on the right that also approaches height \(2\) as \(x\to 3^+\). At \(x=3\), the graph has a filled point at height \(-1\). Decide whether the two-sided limit exists.
Step 1: Read the left-hand behavior. Tracing the left branch toward \(x=3\), its height approaches \(2\). Therefore, \(\lim_{x\to 3^-}f(x)=2\).
Step 2: Read the right-hand behavior. Tracing the right branch toward \(x=3\), its height also approaches \(2\). Therefore, \(\lim_{x\to 3^+}f(x)=2\).
Step 3: Compare. Both one-sided limits are finite and equal, so the two-sided limit exists and equals \(2\).
Check: The filled point says \(f(3)=-1\). The limit concerns nearby inputs with \(x\ne3\), so that function value does not change the conclusion. The two approaching heights are both \(2\), as required.
Worked Example: Different Heights on the Two Sides
Suppose a graph approaches height \(5\) as \(x\to -2^-\) and height \(1\) as \(x\to -2^+\). Decide whether \(\lim_{x\to -2}f(x)\) exists.
Step 1: Record the left-hand limit. The left branch approaches \(5\), so \(\lim_{x\to -2^-}f(x)=5\).
Step 2: Record the right-hand limit. The right branch approaches \(1\), so \(\lim_{x\to -2^+}f(x)=1\).
Step 3: Compare the values. Since \(5\ne1\), the one-sided limits do not match. There is no single finite value approached from both sides.
Check: The failure is not caused by the graph having a gap or a filled point at \(x=-2\). It follows directly from the unequal one-sided limits.
Reading the Test from a Table
A table gives sample function values rather than the entire graph. To investigate a two-sided limit, look at inputs below \(c\) that get closer to \(c\), and inputs above \(c\) that get closer to \(c\). Estimate the limiting value from each side separately before comparing them.
Tables provide evidence for a limit, not a proof in general. A finite number of entries cannot show what happens at every input arbitrarily close to \(c\). In particular, rounded data can conceal small differences, and values that appear to settle may later behave differently. State a conclusion as an estimate when the table only supports an estimate.
Worked Example: Table Values Suggesting a Shared Limit
The following values are given for inputs approaching \(1\) from both sides. Estimate whether a finite two-sided limit exists.
| Approach | Input \(x\) | Value \(f(x)\) |
|---|---|---|
| From the left | 0.9 | 3.1 |
| From the left | 0.99 | 3.01 |
| From the left | 0.999 | 3.001 |
| From the right | 1.1 | 2.9 |
| From the right | 1.01 | 2.99 |
| From the right | 1.001 | 2.999 |
Step 1: Compare left-side outputs with \(3\). The differences are \(3.1-3=0.1\), \(3.01-3=0.01\), and \(3.001-3=0.001\). The inputs get closer to \(1\), and the outputs get closer to \(3\).
Step 2: Compare right-side outputs with \(3\). The absolute differences are \(|2.9-3|=0.1\), \(|2.99-3|=0.01\), and \(|2.999-3|=0.001\). These outputs also get closer to \(3\).
Step 3: Compare the side estimates. Both sides suggest the same finite value, \(3\). The table therefore supports the estimate \(\lim_{x\to1}f(x)\approx3\).
Check: On each side the listed distances from \(3\) shrink from \(0.1\) to \(0.01\) to \(0.001\). Both one-sided estimates agree, so \(3\) is the reasonable estimate from these data. The table alone does not establish the behavior at every nearby input.
Worked Example: Table Values Suggesting Unequal Limits
Suppose a table near \(x=0\) gives left-side values \(2.2,\ 2.02,\ 2.002\) at inputs \(-0.1,\ -0.01,\ -0.001\), and right-side values \(4.1,\ 4.01,\ 4.001\) at inputs \(0.1,\ 0.01,\ 0.001\). What does the table suggest?
Step 1: Estimate from the left. The left outputs approach \(2\): their differences from \(2\) are \(0.2,\ 0.02,\ 0.002\). Thus the left-hand limit is estimated as \(2\).
Step 2: Estimate from the right. The right outputs approach \(4\): their differences from \(4\) are \(0.1,\ 0.01,\ 0.001\). Thus the right-hand limit is estimated as \(4\).
Step 3: Apply the criterion. Since \(2\ne4\), the data suggest unequal one-sided limits and therefore no finite two-sided limit.
Check: The differences from the separate estimates shrink on both sides, but the estimates themselves remain different. A shared finite value cannot equal both \(2\) and \(4\).
When a Side Does Not Settle to One Finite Value
The comparison test requires a finite one-sided limit from each direction. If a branch rises without bound on one side, or its values fail to approach any single finite height, it cannot match the other side at a common finite value. A table may suggest this kind of behavior, but sparse data should be treated cautiously.
Movement up and down alone does not show that a limit fails. Oscillations can shrink toward one height. For instance, \(f(x)=x\sin(1/x)\) for \(x\ne0\) oscillates near \(0\), but \(|\sin(1/x)|\le1\), so \(|x\sin(1/x)|\le|x|\). As \(x\) approaches \(0\) from either side, \(|x|\) approaches \(0\), and the function values are forced toward \(0\). The oscillations shrink, and both one-sided limits equal \(0\).
By contrast, oscillations that do not shrink toward a single height provide no single finite one-sided value to compare. The important question is not whether values move up and down, but whether they get arbitrarily close to one fixed value as the input approaches \(c\).
A Common Mistake
A frequent error is to inspect one side, find a plausible value, and conclude that the two-sided limit equals it without checking the other side.
This conclusion does not follow from the left side alone. For example, if the right branch approaches \(6\), the left-hand limit is \(3\) while the right-hand limit is \(6\). Since \(3\ne6\), the two-sided limit does not exist. The correction is to record both one-sided values and apply the equality test only after checking each side.
A Reliable Decision Process
Identify the value \(c\) that \(x\) is approaching.
Use graph behavior or inputs below \(c\) to estimate whether the function approaches one finite value.
Use graph behavior or inputs above \(c\) to estimate the corresponding value independently.
If both finite values exist and are equal, that shared value is the two-sided limit. Otherwise, there is no finite two-sided limit.
Keep the function value at \(c\) separate from this process. A hole, a filled point at a different height, or a missing function value does not decide the limit by itself. The decision depends on the behavior of nearby inputs from both sides.
Check Your Understanding
For each question, decide what the graph or data imply about the finite two-sided limit. Explain your comparison of the one-sided behavior.
- A graph approaches height \(7\) from the left and height \(7\) from the right, but \(f(c)=0\). What is the two-sided limit?
- A graph approaches height \(-1\) from the left and height \(2\) from the right. Does a finite two-sided limit exist?
- A table has outputs approaching \(5\) from below \(c\) and approaching \(5\) from above \(c\). What does this evidence suggest, and what limitation does a finite table have?
- Why is a left-hand limit alone insufficient to determine whether the two-sided limit exists?
- A graph oscillates more and more tightly around height \(0\) near \(c\). Does the oscillation by itself show that the limit fails? Explain.
