Comparing the Two Sides
A function may behave differently just to the left and just to the right of an input. A two-sided limit asks for one value that describes the function's behavior as the input approaches from both sides. The key test is therefore to compare the left-hand and right-hand limits.
The earlier material on one-sided limits introduced the notation \(\lim_{x\to a^-}f(x)\) for an approach from the left and \(\lim_{x\to a^+}f(x)\) for an approach from the right. In this tutorial, we combine those two ideas into a criterion for a two-sided limit.
The comparison concerns what the function approaches near \(a\), not necessarily the value \(f(a)\). The function may have a different value at \(a\), or may not be defined there at all, without changing whether its two-sided limit exists.
What to Look for in a Graph
To estimate a two-sided limit from a graph, follow the curve toward the input from each direction. Ask what height the left branch approaches, then ask what height the right branch approaches. If both approaches have the same height, that height is the candidate for the two-sided limit. If the approaches have different heights, there is no single value that describes both sides.
A point drawn at the input can be useful for identifying \(f(a)\), but it does not replace this comparison. In particular, an open circle or a filled point at a different height does not by itself prevent the two branches from approaching the same value.
Worked Example: Matching Branches with a Different Point Value
Suppose the graph of \(h\) has the following branches near \(x=2\): for \(x<2\), \(h(x)=3+(x-2)\); for \(x>2\), \(h(x)=3-(x-2)\). At \(x=2\), the graph has a filled point at height \(-1\), so \(h(2)=-1\). Estimate \(\lim_{x\to 2}h(x)\).
First approach from the left. At \(x=1.9\), the function value is \(3+(1.9-2)=3-0.1=2.9\). At \(x=1.99\), it is \(3+(1.99-2)=3-0.01=2.99\). These values approach \(3\) as \(x\) approaches \(2\) from the left.
Now approach from the right. At \(x=2.1\), the value is \(3-(2.1-2)=3-0.1=2.9\). At \(x=2.01\), it is \(3-(2.01-2)=3-0.01=2.99\). These also approach \(3\), now as \(x\) approaches \(2\) from the right.
Both sides approach the same height, so the two-sided limit exists and equals \(3\). The filled point gives \(h(2)=-1\), which is different, but that does not change the limit.
Check: The left and right branch formulas both give values approaching \(3\) as their inputs approach \(2\). The separate point value \(-1\) is not used in that check.
What to Look for in a Table
A table provides selected function values rather than a complete picture of the curve. To investigate a two-sided limit, examine inputs below \(a\) and above \(a\), and compare the corresponding trends in the function values. A useful table includes inputs on both sides that get progressively closer to \(a\).
Matching trends in a table suggest a common limit. Mismatching trends suggest that the two-sided limit does not exist. A finite table is evidence, not a general proof: it shows only the listed inputs. When a formula is available, use it to check what happens as inputs continue to approach the target from each side.
Worked Example: A Table with Matching Trends
Suppose that near \(x=0\), the function is defined by \(f(x)=4+x\) for \(x<0\) and \(f(x)=4+2x\) for \(x>0\). The value at \(x=0\) is not needed. Use values on both sides to estimate the two-sided limit.
For the left side, at \(x=-0.1\), \(f(x)=4+(-0.1)=3.9\). At \(x=-0.01\), \(f(x)=4+(-0.01)=3.99\). At \(x=-0.001\), \(f(x)=4+(-0.001)=3.999\). The values approach \(4\) from below.
For the right side, at \(x=0.1\), \(f(x)=4+2(0.1)=4+0.2=4.2\). At \(x=0.01\), \(f(x)=4+2(0.01)=4+0.02=4.02\). At \(x=0.001\), \(f(x)=4+2(0.001)=4+0.002=4.002\). These values approach \(4\) from above.
| Side | Input | Function value |
|---|---|---|
| Left | \(-0.1\) | \(3.9\) |
| Left | \(-0.01\) | \(3.99\) |
| Right | \(0.1\) | \(4.2\) |
| Right | \(0.01\) | \(4.02\) |
The two trends agree on \(4\), so \(\lim_{x\to 0}f(x)=4\). The formula confirms the estimate: on the left, \(4+x\) approaches \(4\) as \(x\) approaches \(0\); on the right, \(4+2x\) also approaches \(4\).
Check: The listed values on each side are consistent with the formula, and both one-sided approaches have the same limiting value, \(4\).
When the Sides Do Not Match
If the left and right sides approach different finite values, the two-sided limit does not exist. The reason is direct: a two-sided limit would have to describe the behavior from both directions with one value. If the left approaches one value and the right another, no single value meets that requirement.
Worked Example: Different Heights on the Two Sides of a Graph
Suppose a graph has a horizontal branch at height \(1\) for inputs just less than \(0\), and a horizontal branch at height \(3\) for inputs just greater than \(0\). The function is defined at \(0\) with \(g(0)=7\). Decide whether \(\lim_{x\to 0}g(x)\) exists.
Moving toward \(0\) from the left, the graph stays at height \(1\). Thus \(\lim_{x\to 0^-}g(x)=1\). Moving toward \(0\) from the right, the graph stays at height \(3\). Thus \(\lim_{x\to 0^+}g(x)=3\).
Since \(1\ne 3\), the two sides do not approach the same value. Therefore, \(\lim_{x\to 0}g(x)\) does not exist. The point value \(g(0)=7\) is a separate fact and cannot make the two branches agree.
Check: The left branch approaches \(1\), while the right branch approaches \(3\). Since those values differ, there is no common two-sided limit.
Worked Example: Different Trends in a Table
Suppose that near \(x=1\), \(q(x)=2\) when \(x<1\) and \(q(x)=5\) when \(x>1\). Use the table to decide whether a two-sided limit exists.
On the left, at \(x=0.9\), \(q(x)=2\); at \(x=0.99\), \(q(x)=2\); and at \(x=0.999\), \(q(x)=2\). The left-hand values stay at \(2\), so the left-hand limit is \(2\).
On the right, at \(x=1.1\), \(q(x)=5\); at \(x=1.01\), \(q(x)=5\); and at \(x=1.001\), \(q(x)=5\). The right-hand values stay at \(5\), so the right-hand limit is \(5\).
| Side | Input | Function value |
|---|---|---|
| Left | \(0.9\) | \(2\) |
| Left | \(0.99\) | \(2\) |
| Right | \(1.1\) | \(5\) |
| Right | \(1.01\) | \(5\) |
The two-sided limit does not exist because the one-sided limits are unequal: \(2\ne 5\). The table illustrates the two trends, and the formulas for \(q\) confirm them for inputs arbitrarily close to \(1\) on each side.
Check: Every listed left-side value matches the left-hand rule, and every listed right-side value matches the right-hand rule. The two sides approach different values, so they cannot give a common limit.
Why the Criterion Works
The criterion follows from what it means to approach an input from both directions. A two-sided approach includes inputs just to the left of \(a\) and inputs just to the right of \(a\). Restricting that behavior to just one side gives the corresponding one-sided approach. Conversely, if both sides get close to the same output value, then inputs close to \(a\) on either side get close to that value.
The function values approach \(L\) for inputs approaching \(a\) from either direction. Restricting to inputs less than \(a\) gives the left-hand limit \(L\); restricting to inputs greater than \(a\) gives the right-hand limit \(L\).
For any chosen degree of closeness to \(L\), the left-hand limit ensures that values are that close when inputs are sufficiently near \(a\) from the left. The right-hand limit ensures the same for inputs sufficiently near from the right.
Use the shorter of the two left- and right-side intervals. Every input in this combined neighborhood, except \(a\) itself, lies on one side or the other. On either side, the function value is close to \(L\). Therefore the two-sided limit is \(L\).
This argument also explains why matching matters. Both one-sided limits must not merely exist; they must be equal. If they differ, the combined neighborhood cannot make the function approach one common value from both directions.
A Common Mistake: Using the Point Value
A frequent error is to treat the value at the input as if it determined the limit. For instance, in the first worked example, one might reason incorrectly: “Since \(h(2)=-1\), the limit is \(-1\).” That conclusion confuses the function's value at \(2\) with the behavior of nearby values.
The correction is to compare the two sides. The left and right branches both approach \(3\), so the correct conclusion is \(\lim_{x\to 2}h(x)=3\), even though \(h(2)=-1\).
Another useful check is to write down the two one-sided conclusions before writing a two-sided limit. If they match, state their common value. If they do not match, state that the two-sided limit does not exist. Do not average two different one-sided values: an average is not the value approached by either side.
Using the Test Carefully
Graphs and tables help estimate one-sided behavior, but their evidence has limits. A graph can be drawn at a misleading scale, and a table contains only finitely many inputs. When possible, use the function's formula or a clear description of its branches to verify what happens as inputs move closer to the target.
For this comparison, the essential questions are:
- What value does the function approach from the left?
- What value does it approach from the right?
- Are both one-sided limits defined and equal?
If the answers give a shared value \(L\), the two-sided limit exists and equals \(L\). If the values differ, there is no two-sided limit. The method applies regardless of the function's value at the target input.
Check Your Understanding
For each question, compare the behavior on both sides of the target input.
- A graph's left and right branches both approach height \(6\), while the filled point at the target has height \(2\). What is the two-sided limit?
- A table suggests that values from the left approach \(4\), while values from the right approach \(4\). What additional information would make this evidence more convincing?
- The left-hand limit at \(x=a\) is \(-2\), and the right-hand limit is \(1\). Does the two-sided limit exist? Explain.
- Why does the value \(f(a)\) not determine whether \(\lim_{x\to a}f(x)\) exists?
- State the criterion for \(\lim_{x\to a}f(x)=L\) in terms of the two one-sided limits.
