Reading Behavior at a Piecewise Boundary
A piecewise function is defined by different formulas on different parts of its input domain. At a point where the formula changes, evaluating the function can require a little care: the formula used at the boundary may not be the formula used for nearby inputs on both sides.
To estimate a limit at such a boundary, the key question is not only, “What is the function’s value there?” Instead, examine inputs close to the boundary, select the formula that applies to each input, and see what output values those calculations suggest.
This tutorial focuses on numerical estimates and the reasoning behind them. A table can help reveal a trend, but a few computed values alone do not prove what happens for every input sufficiently close to the boundary.
A Reliable Method
Suppose a piecewise definition changes at \(x=c\). First identify precisely which formula applies when \(x<c\), which applies when \(x>c\), and which formula (if any) defines \(f(c)\). Then choose inputs close to \(c\) on either side. The inputs must be close enough to show local behavior but distinct from \(c\), so that the nearby branch formulas are actually being used.
Find the input where the piecewise conditions switch from one formula to another.
Check whether each condition includes the boundary itself, for example \(x\le c\), or excludes it, for example \(x<c\).
Use a few inputs on each side of the boundary, including some closer inputs for a check.
Substitute each input into its own branch, not into a formula chosen by guesswork.
Use the nearby outputs to estimate the trend, and calculate \(f(c)\) separately.
When choosing table entries, do not accidentally include \(c\) among the inputs meant to describe nearby behavior. The value at \(c\) can be useful to report, but it is not a substitute for evaluating inputs close to \(c\).
Worked Example: Two Branches Meet at the Boundary
Worked Example: Two Branches Meet at the Boundary
Let
Estimate the limit at the boundary \(x=0\), and find the function value \(f(0)\).
Step 1: Identify the formulas. For inputs less than \(0\), use \(x^2+1\). For inputs greater than or equal to \(0\), use \(1-2x\). In particular, the second formula defines the function value at \(0\).
Step 2: Evaluate nearby inputs. On the negative side, at \(x=-0.1\), \(f(-0.1)=(-0.1)^2+1=0.01+1=1.01\). At \(x=-0.01\), \(f(-0.01)=(-0.01)^2+1=0.0001+1=1.0001\).
On the positive side, at \(x=0.01\), \(f(0.01)=1-2(0.01)=1-0.02=0.98\). At \(x=0.1\), \(f(0.1)=1-2(0.1)=1-0.2=0.8\).
Step 3: Estimate and check. The outputs at inputs closer to \(0\) are closer to \(1\) on both sides, so the table suggests the limit is \(1\). Direct substitution into the branch that includes \(0\) gives \(f(0)=1-2(0)=1\).
As a check with closer inputs, \(f(-0.001)=(-0.001)^2+1=0.000001+1=1.000001\), and \(f(0.001)=1-2(0.001)=0.998\). These values are again close to \(1\). This check supports the estimate; the algebraic differences discussed below explain the trend more fully.
Why the Nearby Values Move Toward the Estimate
A table makes a trend visible, but it is helpful to see why the calculations in the example support the estimate. On the negative side, subtracting the proposed value \(1\) from the output gives
Thus, for negative inputs close to zero, the output differs from \(1\) by the square of the input. For instance, when \(x=-0.01\), the difference is \((-0.01)^2=0.0001\), which matches the computed output \(1.0001\).
On the positive side, the corresponding difference is
At \(x=0.01\), this difference is \(-2(0.01)=-0.02\), so the output is \(1-0.02=0.98\). As positive inputs are chosen closer to zero, this difference becomes smaller in size. The two branch calculations therefore explain why outputs on both sides are near \(1\) for inputs near the boundary.
This kind of calculation is more informative than a list of values alone: it relates the output’s difference from the proposed estimate to the input’s distance from the boundary. The table helps us notice the pattern; the formulas explain it.
Worked Example: The Boundary Value Is Different
Worked Example: The Boundary Value Is Different
Define
Estimate the limit at \(x=2\), and compare it with \(g(2)\).
Step 1: Separate the boundary from nearby inputs. The definition assigns \(g(2)=10\). Inputs less than \(2\) use \(2x-1\), while inputs greater than \(2\) use \(x+1\).
Step 2: Calculate on the lower side. For \(x=1.9\), \(g(1.9)=2(1.9)-1=3.8-1=2.8\). For \(x=1.99\), \(g(1.99)=2(1.99)-1=3.98-1=2.98\).
Step 3: Calculate on the upper side. For \(x=2.01\), \(g(2.01)=2.01+1=3.01\). For \(x=2.1\), \(g(2.1)=2.1+1=3.1\).
Step 4: Estimate and check. The closer inputs, \(1.99\) and \(2.01\), give outputs \(2.98\) and \(3.01\), both close to \(3\). The table suggests that the limit is \(3\), even though \(g(2)=10\). With closer inputs, \(g(1.999)=2(1.999)-1=3.998-1=2.998\), and \(g(2.001)=2.001+1=3.001\), which supports the same estimate.
This example makes the distinction important: the function has value \(10\) at the boundary, while nearby inputs produce values close to \(3\). The boundary value and the estimated limit answer different questions.
Worked Example: The Boundary Belongs to One Branch
Worked Example: The Boundary Belongs to One Branch
Let
Estimate the limit at \(x=-1\), and calculate \(h(-1)\).
Step 1: Read the inequalities carefully. The first formula includes \(x=-1\), so \(h(-1)=2(-1)+5=-2+5=3\). Inputs greater than \(-1\), however, use the second formula.
Step 2: Evaluate inputs below the boundary. At \(x=-1.1\), \(h(-1.1)=2(-1.1)+5=-2.2+5=2.8\). At \(x=-1.01\), \(h(-1.01)=2(-1.01)+5=-2.02+5=2.98\).
Step 3: Evaluate inputs above the boundary. At \(x=-0.99\), \(h(-0.99)=(-0.99)^2+2=0.9801+2=2.9801\). At \(x=-0.9\), \(h(-0.9)=(-0.9)^2+2=0.81+2=2.81\).
Step 4: Estimate and check. The outputs from inputs close to \(-1\) are close to \(3\), suggesting a limit of \(3\). For a closer check, \(h(-1.001)=2(-1.001)+5=-2.002+5=2.998\), while \(h(-0.999)=(-0.999)^2+2=0.998001+2=2.998001\). Both are close to \(3\), and the separate boundary calculation also gives \(h(-1)=3\).
A Common Mistake: Using the Boundary Value as the Limit
A tempting shortcut is to find \(f(c)\) and assume it must be the limit at \(c\). That shortcut fails in the second worked example. The incorrect calculation would be:
The conclusion does not follow. The calculation finds the output at the single input \(2\); it does not examine nearby inputs. In fact, the branch formulas give \(g(1.99)=2.98\) and \(g(2.01)=3.01\), which support an estimate of \(3\), not \(10\).
A second common error is to choose a formula based on which output seems plausible rather than on the input condition. For example, in the second example, \(x=2.01\) satisfies \(x>2\), so its value must be computed using \(x+1\). Using \(2x-1\) there would calculate an expression that does not define \(g(2.01)\).
What a Table Can and Cannot Tell You
Tables are useful evidence because they let you compare outputs for inputs increasingly close to the boundary. A good table uses both sides when both sides have nearby inputs in the domain, and it uses the correct formula for every entry. If a trend is unclear, choose closer inputs or inspect how the relevant formula differs from the proposed output value.
However, a finite table checks only finitely many inputs. It supports an estimate but does not by itself prove that the same behavior continues for every input arbitrarily close to the boundary. To justify a limit fully, one must use the behavior of the formulas, not only a handful of decimal calculations. The difference calculations for the first example illustrate how the formulas can explain a numerical trend.
Whenever a piecewise definition is involved, keep three records separate: the formula for inputs below the boundary, the formula for inputs above the boundary, and the actual value assigned at the boundary. This prevents a single point from being confused with the behavior indicated by nearby inputs.
Check Your Understanding
Use the branch conditions carefully, and distinguish estimates from function values.
- For \(p(x)=\begin{cases}x+4,&x<1,\\2x+3,&x\ge1,\end{cases}\) which formula applies at \(x=0.99\), and which applies at \(x=1\)?
- For the function \(g\) in the second worked example, what value does the definition assign at \(x=2\), and what limit does the nearby-value table suggest?
- Why must inputs on either side of a boundary be evaluated using their own branch formulas?
- Give one reason why a table with only a few entries cannot, by itself, prove a limit value.
- For \(h\) in the third worked example, which branch gives \(h(-1)\), and why does the other branch apply to inputs just greater than \(-1\)?
