Introduction
When studying what a function does near an input \(a\), it matters not only that the inputs get close to \(a\), but also which side they come from. Inputs less than \(a\) approach from the left; inputs greater than \(a\) approach from the right.
Graphs and tables can show these two directions separately. The task is to track the inputs as they get closer to the target, then observe the corresponding function values. The function’s value at the target itself may be different from either nearby trend.
Two Directions of Approach
For example, if \(a=3\), then \(x=2\) satisfies \(x<a\), but by itself that fact does not mean \(x\) is approaching \(3\). A sequence such as \(2.9, 2.99, 2.999\) consists of inputs below \(3\) that get closer to it. Likewise, \(3.1, 3.01, 3.001\) are above \(3\) and get closer from the right.
The standard notation records the direction:
The superscript minus or plus indicates the side of the input, not whether the function values are negative or positive. For instance, a left-hand approach can have positive function values, and a right-hand approach can have negative function values.
Reading a Table One Direction at a Time
A table is useful only if its inputs are chosen on the side being investigated and get progressively closer to the target. Read the input column first: decide whether each input is below or above \(a\). Then follow the corresponding function values and look for the trend.
A few entries can suggest a trend, but a table is a sample of values, not a proof by itself. When a function formula is available, it can confirm exactly what the values do as the inputs approach from a given side.
Worked Example: Compare the Two Directions in a Table
Define \(f\) near \(x=1\) by
First examine inputs below \(1\). For \(x=0.9\), \(f(x)=0.9+2=2.9\). For \(x=0.99\), \(f(x)=0.99+2=2.99\). For \(x=0.999\), \(f(x)=0.999+2=2.999\).
| Input \(x<1\) | Calculation | Output \(f(x)\) |
|---|---|---|
| 0.9 | 0.9 + 2 | 2.9 |
| 0.99 | 0.99 + 2 | 2.99 |
| 0.999 | 0.999 + 2 | 2.999 |
These inputs approach \(1\) from below, and the outputs approach \(3\). The formula confirms the trend: for \(x<1\), \(f(x)=x+2\), so \(f(x)-3=x-1\). As \(x\) gets closer to \(1\), the difference \(x-1\) gets closer to \(0\).
Now take inputs above \(1\). The rule for every \(x>1\) is \(f(x)=4\), so each calculation gives \(4\): \(f(1.1)=4\), \(f(1.01)=4\), and \(f(1.001)=4\).
| Input \(x>1\) | Calculation | Output \(f(x)\) |
|---|---|---|
| 1.1 | Rule for \(x>1\) | 4 |
| 1.01 | Rule for \(x>1\) | 4 |
| 1.001 | Rule for \(x>1\) | 4 |
Thus the table and the rules show that the left-hand values approach \(3\), while the right-hand values stay at \(4\). The separately specified value \(f(1)=-2\) does not change either calculation: none of the inputs in these two approaches equals \(1\).
Check: Substituting \(x=0.999\) into the left rule gives \(0.999+2=2.999\), as in the table. Substituting \(x=1.001>1\) into the right rule gives \(4\), also as in the table.
Reading a Graph from One Side
To read a graph, locate the target input \(a\) on the horizontal axis. For a left-hand approach, trace the curve toward \(x=a\) while moving from inputs less than \(a\). For a right-hand approach, trace it toward \(x=a\) from inputs greater than \(a\). Watch the height of the curve: that height is the function value.
An open circle marks a point not included in the graph at that location. A filled point marks a function value there. Either can appear at the target input, but neither replaces the task of tracing nearby values from the specified side. In particular, the height of a filled point at \(a\) need not match the height the curve approaches.
Worked Example: Read a Left-Hand Trend from a Graph
Suppose a graph follows the line \(y=2x-1\) for inputs \(x<3\). The line ends in an open circle at \((3,5)\), while the graph also has a filled point at \((3,1)\). Read the left-hand behavior at \(a=3\).
First identify the direction: from the left, the inputs must satisfy \(x<3\) and get closer to \(3\). Along the branch, the graph’s height is \(2x-1\). At \(x=2.9\), the height is \(2(2.9)-1=5.8-1=4.8\). At \(x=2.99\), it is \(2(2.99)-1=5.98-1=4.98\). At \(x=2.999\), it is \(2(2.999)-1=5.998-1=4.998\).
| Input \(x<3\) | Height \(2x-1\) |
|---|---|
| 2.9 | 4.8 |
| 2.99 | 4.98 |
| 2.999 | 4.998 |
The heights approach \(5\), matching the open circle at \((3,5)\). The filled point at \((3,1)\) gives the value at the target, \(f(3)=1\), but the curve traced from inputs below \(3\) approaches height \(5\).
Check: Using \(x=2.999\) in the branch formula gives \(2(2.999)-1=5.998-1=4.998\), which agrees with the table and lies close to \(5\).
Why a Linear Branch Has the Trend Its Graph Shows
The connection between a graph’s height and its limiting trend can be justified directly for a line. Consider the branch \(f(x)=2x+1\) for inputs \(x<3\). Its height at \(x=3\), if the line formula is extended to that input, is \(2(3)+1=7\). For a nearby input below \(3\), compare the actual height with \(7\):
Because \(x<3\), the difference \(x-3\) is negative, so \(f(x)<7\). Its distance from \(7\) is
As the input comes closer to \(3\) from below, the positive gap \(3-x\) becomes smaller, and the output gap is exactly twice that amount. For example, if \(x=2.99\), then \(3-x=0.01\), so \(|f(x)-7|=2(0.01)=0.02\), and \(f(2.99)=6.98\). If \(x=2.999\), then \(3-x=0.001\), so \(|f(x)-7|=0.002\), and \(f(2.999)=6.998\). This calculation explains why the graph’s heights approach \(7\) from below.
The same reasoning applies to a right-hand approach when the branch is defined for \(x>3\): then \(x-3\) is positive, and the output difference \(2(x-3)\) is positive. The direction of the inputs and the corresponding change in output must be tracked separately.
When the Domain Supplies Only One Side
A function may not have domain inputs on both sides of a target. In that case, only the available direction can be read from nearby function values. For example, the square-root expression \(\sqrt{x-2}\) is defined for real \(x\geq2\), so inputs in its domain near \(2\), other than \(2\) itself, come from the right. The domain itself tells us not to look for values just below \(2\).
Worked Example: Read a Right-Hand Trend from a Table
Let \(g(x)=\sqrt{x-2}\) for \(x\geq2\), and examine inputs approaching \(2\) from the right. Choose \(x=2.01\), \(2.0001\), and \(2.000001\). Each is greater than \(2\), so each is on the required side.
For \(x=2.01\), \(x-2=0.01\), and \(g(x)=\sqrt{0.01}=0.1\). For \(x=2.0001\), \(x-2=0.0001\), and \(g(x)=\sqrt{0.0001}=0.01\). For \(x=2.000001\), \(x-2=0.000001\), and \(g(x)=\sqrt{0.000001}=0.001\).
| Input \(x>2\) | Input difference \(x-2\) | Output \(\sqrt{x-2}\) |
|---|---|---|
| 2.01 | 0.01 | 0.1 |
| 2.0001 | 0.0001 | 0.01 |
| 2.000001 | 0.000001 | 0.001 |
The outputs get closer to \(0\) as the right-side inputs get closer to \(2\). The calculations can be checked by squaring: \(0.1^2=0.01\), \(0.01^2=0.0001\), and \(0.001^2=0.000001\). These are exactly the corresponding values of \(x-2\), so the displayed square roots are correct.
The function is defined at the target, and \(g(2)=\sqrt{0}=0\). Here that value agrees with the observed trend, but the direction of approach is established by the nearby inputs, not merely by evaluating at \(2\).
A Common Mistake: Using the Target Value Instead of the Trend
A frequent error is to read the function value at the target and report it as the limiting value without tracing the graph or checking nearby inputs. Consider the graph in the earlier example: it has a filled point at \((3,1)\), but the left branch approaches height \(5\).
Incorrect calculation: “The filled point is at height \(1\), so the left-hand behavior is \(1\).” This uses \(f(3)=1\), even though a left-hand approach uses inputs \(x<3\), not \(x=3\).
Correction: Trace the branch for \(x<3\). Its formula gives \(f(x)=2x-1\), and the nearby heights \(4.8, 4.98, 4.998\) approach \(5\). The point value and the nearby trend are different pieces of information; report each according to what is being asked.
Another related mistake is treating \(x<a\) alone as a complete description of approaching from the left. For \(a=3\), the input \(x=1\) is below \(3\), but one isolated input does not show a pattern of getting arbitrarily close. A useful graph trace or table must both stay on the selected side and move closer to the target.
A Reliable Reading Routine
Mark \(a\) on the horizontal axis or identify it in the table.
For a left-hand approach use domain inputs \(x<a\); for a right-hand approach use domain inputs \(x>a\).
A point on the correct side is not enough; the inputs must move arbitrarily close to \(a\).
Follow graph heights or table outputs to determine the value they approach.
Record \(f(a)\), if defined, separately from the trend of nearby values.
Why Direction Matters
Looking at only one side can reveal behavior that a broad view hides. A branch might rise as it approaches a target from the left and behave differently on the right; a domain may provide inputs on just one side; or a graph may have an isolated filled point that does not describe either nearby trend. Careful side-by-side tracking prevents these features from being confused.
When reading a graph, follow the curve toward the target without jumping to a filled point. When reading a table, sort the inputs by side before interpreting the outputs. In both cases, the central question is precise: as inputs on this particular side get closer to \(a\), what value do the corresponding outputs approach?
Check Your Understanding
For each question, identify the side of approach before deciding what the outputs indicate.
- If \(a=4\), which of the inputs \(3.9\), \(4.01\), and \(3\) is below the target and close to it? Which is above it and close to it?
- A graph has an open circle at \((2,6)\) and a filled point at \((2,0)\). The branch for \(x<2\) moves toward the open circle. What height does the branch approach from the left, and what is the function value at \(2\)?
- For \(h(x)=x+5\) when \(x<1\), calculate \(h(0.9)\) and \(h(0.99)\). What trend do these outputs suggest as inputs approach \(1\) from below?
- Why does the input \(x=1\) by itself not show that inputs are approaching \(a=3\) from the left, even though \(1<3\)?
- The domain of \(p(x)=\sqrt{x+1}\) is \(x\geq-1\). Near the target \(a=-1\), which side supplies domain inputs other than the target?
