Why Check Both Sides?
A function can behave differently just to the left and just to the right of an input. A graph may approach one height from the left and another height from the right; a table may show values tending toward two different numbers. In either case, a single two-sided limit cannot describe both trends.
To diagnose a two-sided limit, examine the function as the input approaches \(a\) from each side separately. The function’s value at \(a\) is not the deciding factor: the question concerns nearby inputs, not necessarily the value at the input itself.
One-Sided Limits
Let \(f\) be a function with domain containing inputs arbitrarily close to \(a\) on the relevant side. The statement \(\lim_{x\to a^-}f(x)=L\) means that the values of \(f(x)\) can be made as close as desired to \(L\) by taking \(x<a\) sufficiently close to \(a\). The notation \(a^-\) indicates approach from the left.
Precisely, \(\lim_{x\to a^-}f(x)=L\) if, for every \(\varepsilon>0\), there is a \(\delta>0\) such that for every \(x\) in the domain,
The statement \(\lim_{x\to a^+}f(x)=L\) uses inputs to the right of \(a\): for every \(\varepsilon>0\), there is a \(\delta>0\) such that
The two-sided limit \(\lim_{x\to a}f(x)=L\) requires the same closeness condition for all domain inputs sufficiently near \(a\), on either side, but excludes \(x=a\). These definitions describe trends near \(a\); none requires \(f(a)=L\), or even that \(f(a)\) be defined.
The Two-Sided Limit Test
The reason is built into the definitions. A two-sided approach includes inputs from both directions, so the function must satisfy the required closeness condition on each side. Conversely, if each side satisfies that condition for the same \(L\), choose a distance that works on both sides.
For any \(\varepsilon>0\), its definition supplies a \(\delta>0\) that works for all nearby inputs with \(0<|x-a|<\delta\). Restricting to \(x<a\) gives the left-hand condition; restricting to \(x>a\) gives the right-hand condition. Thus both one-sided limits equal \(L\).
Fix any \(\varepsilon>0\). The left-hand definition supplies a distance \(\delta_1>0\), and the right-hand definition supplies a distance \(\delta_2>0\). Set \(\delta=\min(\delta_1,\delta_2)\). If \(0<|x-a|<\delta\), then either \(x<a\), when the left-hand condition applies, or \(x>a\), when the right-hand condition applies. In either case, \(|f(x)-L|<\varepsilon\). This is exactly the two-sided limit condition.
This criterion is a diagnosis tool, not a way to average the one-sided answers. They must agree on the same finite value.
Worked Example: A Graph Approaches the Same Height on Both Sides
Worked Example: A Graph Approaches the Same Height on Both Sides
Suppose a graph near \(x=2\) has a curve that approaches height \(5\) as \(x\) approaches \(2\) from the left, and also approaches height \(5\) from the right. The plotted point at \(x=2\) is at height \(1\). Diagnose the two-sided limit.
The left-hand trend is \(5\), so \(\lim_{x\to 2^-}f(x)=5\) according to the graph. The right-hand trend is also \(5\), so \(\lim_{x\to 2^+}f(x)=5\). The one-sided values agree. Therefore the graph supports the conclusion
The point at height \(1\) tells us \(f(2)=1\), but it does not change the limit. Check: approaching from either side gives the same height, \(5\), as required by the two-sided limit test. A graph provides visual evidence; without additional information about the function, the drawing alone is not a formal proof.
Worked Example: A Table Suggests Matching Trends
Worked Example: A Table Suggests Matching Trends
Consider \(g(x)=3+(x-1)^2\) for \(x\ne 1\). Use nearby values to assess the limit as \(x\) approaches \(1\).
From the left, take \(x=0.9\), \(0.99\), and \(0.999\). The differences from \(1\) are \(-0.1\), \(-0.01\), and \(-0.001\). Squaring gives \(0.01\), \(0.0001\), and \(0.000001\). Thus the function values are \(3+0.01=3.01\), \(3+0.0001=3.0001\), and \(3+0.000001=3.000001\).
From the right, take \(x=1.1\), \(1.01\), and \(1.001\). The differences from \(1\) are \(0.1\), \(0.01\), and \(0.001\). Their squares are again \(0.01\), \(0.0001\), and \(0.000001\), so the function values are \(3.01\), \(3.0001\), and \(3.000001\).
On both sides, the values approach \(3\). The table suggests that the one-sided limits agree, and therefore suggests \(\lim_{x\to 1}g(x)=3\). Here the formula confirms the conclusion: since \((x-1)^2\) approaches \(0\) as \(x\) approaches \(1\), \(3+(x-1)^2\) approaches \(3\). Check: substituting \(x=1\) into the expression gives \(3+(1-1)^2=3+0=3\), consistent with the observed trend. The table alone, with only finitely many entries, would be evidence rather than proof.
Worked Example: Different Trends Rule Out a Two-Sided Limit
Worked Example: Different Trends Rule Out a Two-Sided Limit
Suppose a graph near \(x=0\) approaches height \(-2\) from the left and height \(4\) from the right. Decide whether a finite two-sided limit exists.
The left-hand limit is \(-2\), while the right-hand limit is \(4\). Since \(-2\ne 4\), the one-sided limits do not agree. The two-sided limit test therefore gives
Check: any proposed two-sided limit would have to equal the left-hand limit, hence \(-2\), and also equal the right-hand limit, hence \(4\). No single number can satisfy both requirements. The conclusion does not depend on the value of \(f(0)\).
How to Read Graphs and Tables Carefully
For a graph, follow the curve toward the input from each side. Do not use the height of a plotted point at the input as a substitute for the nearby trend. An open circle, a filled point, or no plotted point at \(a\) can coexist with a limit; what matters is how the graph behaves as \(x\) gets close to \(a\).
For a table, sort the entries into inputs less than \(a\) and inputs greater than \(a\). Check whether the corresponding function values approach the same candidate number. Values should be considered in order of their distance from \(a\), not merely in the order they happen to appear.
A Common Mistake: Averaging the One-Sided Limits
Suppose the left-hand limit is \(2\) and the right-hand limit is \(6\). A tempting but incorrect calculation is to average them:
This does not satisfy the definition. From the left, function values approach \(2\), not \(4\); from the right, they approach \(6\), not \(4\). The correct diagnosis is that the one-sided limits differ, so the two-sided limit does not exist.
Another mistake is to conclude that a limit fails to exist merely because \(f(a)\) is missing or has an unexpected value. That confuses a function’s value at one input with its behavior near that input. Always compare the left and right trends first.
A Practical Diagnosis Routine
When given a graph or a table, use the following sequence:
Identify the value \(a\) that \(x\) approaches.
Follow graph points or table values for \(x<a\), and record the value they approach, if one is evident.
Repeat using inputs \(x>a\).
If both trends approach the same finite number, the evidence supports that two-sided limit. If they approach different values, the two-sided limit does not exist. State whether the conclusion is an estimate from a display or an exact result established from the function.
Why This Diagnosis Matters
A two-sided limit describes a single local trend. It is useful only when that trend is coherent from both directions. Checking the two sides separately prevents a graph’s special point, a misleading table entry, or a numerical average from hiding a failure of agreement.
The key habit is simple: do not decide from one side alone. Establish what happens from the left, establish what happens from the right, and then compare. Matching one-sided trends support a two-sided limit; differing one-sided limits rule it out.
Check Your Understanding
For each question, explain how the left-hand and right-hand behavior determines the conclusion.
- A graph approaches height \(3\) from both sides of \(x=1\), but its plotted point at \(x=1\) has height \(-2\). What does the graph suggest for the two-sided limit, and why?
- A table suggests a left-hand limit of \(0\) and a right-hand limit of \(5\). What is the correct two-sided diagnosis?
- State the precise condition on the two one-sided limits for a finite two-sided limit to exist.
- Why do finitely many table entries provide evidence for a limit but not, by themselves, a proof?
- Can a function have a two-sided limit at an input where its value is different from that limit? Explain.
