When the Two Sides Approach Different Values
A two-sided limit asks what value a function's outputs approach as the input approaches a particular number from either direction. Sometimes the function has one trend on the left and a different trend on the right. When those trends approach different numbers, there is no single value that describes the behavior from both sides.
This is a basic but important way for a two-sided limit to fail. The function may still be defined at the input, and each one-sided limit may exist. The failure comes specifically from the two one-sided limits disagreeing.
The superscript minus or plus indicates the direction of approach; it is not an instruction to subtract or add. In particular, these limits concern inputs near \(a\), not necessarily the value \(f(a)\).
The Test for a Two-Sided Limit
A two-sided limit exists exactly when both one-sided limits exist and are equal. If either side has no limit, or if the two sides approach different values, the two-sided limit does not exist.
Here is why the equality condition is necessary and sufficient. If a two-sided limit is \(L\), then approaching \(a\) from only the left or only the right must still give \(L\): each one-sided approach uses inputs included in the two-sided approach. Thus both one-sided limits equal \(L\). Conversely, if both one-sided limits equal the same value \(L\), then inputs sufficiently near \(a\) on the left produce values near \(L\), and inputs sufficiently near \(a\) on the right do too. These two sides together account for all inputs near \(a\), except \(a\) itself, so the two-sided limit is \(L\). Therefore, if the one-sided limits are unequal, a two-sided limit cannot exist.
This criterion turns the diagnosis into a direct comparison: calculate or read the left-hand trend, calculate or read the right-hand trend, and check whether the resulting values match.
Reading Unequal Trends From a Step Graph
Imagine a graph with a horizontal segment at height \(2\) for every \(x<3\), and a horizontal segment at height \(-1\) for every \(x>3\). At \(x=3\), suppose the graph has a filled point at height \(4\). The left segment approaches height \(2\), while the right segment approaches height \(-1\). The filled point records \(f(3)=4\), but it does not change either approaching trend.
On a step graph, focus on the nearby pieces of the graph on each side of the input. An open circle can indicate that a particular endpoint is not included in a piece; it does not prevent the graph from approaching that circle's height. A filled point tells you the function's value at that input. These are distinct pieces of information.
Worked Example: A Step Graph With Different Side Heights
Suppose a step graph consists of \(f(x)=2\) for \(x<3\), \(f(x)=-1\) for \(x>3\), and a filled point at \((3,4)\). Determine the one-sided limits and whether the two-sided limit exists.
For inputs to the left of \(3\), the graph stays at height \(2\), so its output does not change as the input approaches \(3\). Therefore, \(\lim_{x\to 3^-}f(x)=2\).
For inputs to the right of \(3\), the graph stays at height \(-1\). Therefore, \(\lim_{x\to 3^+}f(x)=-1\).
Since \(2\ne -1\), the one-sided limits disagree. The criterion above gives \(\lim_{x\to 3}f(x)\) does not exist. The filled point separately gives \(f(3)=4\).
Check: From the left, all nearby graph points have height \(2\); from the right, all nearby graph points have height \(-1\). Neither side approaches the filled-point height \(4\), so the graph confirms that the function value does not settle the two-sided-limit question.
Computing the Trends From Piecewise Formulas
A piecewise formula specifies a function by using different expressions on different parts of its domain. Near a point where the expression changes, use the expression that applies on the left to find the left-hand limit and the expression that applies on the right to find the right-hand limit. Do not use a branch outside the interval where it is specified.
Worked Example: Comparing Two Affine Branches
Let
Find the one-sided limits at \(x=2\) and decide whether the two-sided limit exists.
For \(x<2\), use the first branch. As \(x\) approaches \(2\) from the left, \(2x+1\) approaches the value obtained at \(2\):
Thus \(\lim_{x\to 2^-}f(x)=5\). For \(x>2\), the applicable branch is \(x-1\), so as \(x\) approaches \(2\) from the right,
Thus \(\lim_{x\to 2^+}f(x)=1\). Since \(5\ne 1\), \(\lim_{x\to 2}f(x)\) does not exist. The second branch includes \(x=2\), so \(f(2)=2-1=1\); that value is not the left-hand limit.
Check: For example, at \(x=1.9\), the left branch gives \(2(1.9)+1=4.8\), near \(5\). At \(x=2.1\), the right branch gives \(2.1-1=1.1\), near \(1\). These nearby outputs confirm the two different trends.
Worked Example: A Polynomial Branch and a Constant Branch
Consider
For inputs approaching \(2\) from the left, use \(x^2+1\). As \(x\) approaches \(2\), \(x^2\) approaches \(2^2=4\), so the left-hand trend is
For inputs approaching from the right, the function is constantly \(3\), so its outputs remain \(3\). Therefore, \(\lim_{x\to 2^+}g(x)=3\). Since \(5\ne3\), the two-sided limit at \(2\) does not exist. In this example, \(g(2)=3\), because the constant branch includes \(2\).
Check: At \(x=1.9\), the left branch gives \(1.9^2+1=3.61+1=4.61\), which is approaching \(5\) as \(x\) moves toward \(2\). At \(x=2.1\), the right branch still gives \(3\). The nearby values match the calculated one-sided trends.
A Common Mistake: Averaging the Two Values
When the one-sided limits disagree, averaging them does not produce a two-sided limit. A limit must describe the trend from both sides, not a compromise between the two trends.
Worked Example: Why the Average Is Not the Limit
Suppose \(\lim_{x\to 0^-}h(x)=4\) and \(\lim_{x\to 0^+}h(x)=10\). A mistaken calculation might be:
The arithmetic average is \(7\), but the left-side outputs approach \(4\), not \(7\), and the right-side outputs approach \(10\), not \(7\). Since \(4\ne10\), the correct conclusion is that \(\lim_{x\to 0}h(x)\) does not exist.
Check: The two-sided-limit criterion requires equal one-sided limits. Substituting the actual values gives \(4\ne10\), so no single value passes that test. The average is not an additional trend from either side.
Another frequent error is to use \(f(a)\) as the limit simply because the function is defined at \(a\). The function value can differ from both one-sided limits, as the step-graph example showed. Always determine the two trends from nearby inputs first; record the function value separately if it is relevant.
A Reliable Comparison Process
Locate the value \(a\) that the input approaches and determine which formula or graph segment applies on each side.
Use inputs less than \(a\) to determine \(\lim_{x\to a^-}f(x)\).
Use inputs greater than \(a\) to determine \(\lim_{x\to a^+}f(x)\).
If both one-sided limits exist and are equal, that common value is the two-sided limit. If they are unequal, the two-sided limit does not exist.
Compute \(f(a)\) from the branch that includes \(a\), if one does. This value does not replace the one-sided-limit comparison.
Why This Diagnosis Matters
Recognizing unequal one-sided trends prevents a misleading single answer from being assigned to a function's behavior. It also makes graph and piecewise-formula questions systematic: the left and right are checked separately, then compared. A discontinuity in the graph or a change of formula may draw attention to a point, but the decisive evidence for this particular failure is the mismatch between the two approaching values.
Keep the conclusion precise. When the side limits are \(L_-\) and \(L_+\) with \(L_-\ne L_+\), say that the two-sided limit does not exist because the one-sided limits are unequal. Do not say that the limit is their average, or that it equals the function's value at the point unless the side comparison independently supports that value.
Check Your Understanding
For each question, compare the behavior from the left and from the right before deciding whether a two-sided limit exists.
- A step graph has height \(6\) just to the left of \(x=1\) and height \(6\) just to the right. What is the two-sided limit?
- A piecewise function has left-hand limit \(-2\) and right-hand limit \(5\) at \(x=4\). Does its two-sided limit exist? Explain.
- For \(p(x)=x+2\) when \(x<1\), and \(p(x)=0\) when \(x\ge1\), find both one-sided limits at \(1\). Does the two-sided limit exist?
- If \(q(3)=8\), must either one-sided limit at \(3\) equal \(8\)? Explain why or why not.
- A student averages unequal one-sided limits and calls the result the two-sided limit. What condition has the student failed to check?
