Why the Domain Determines the Direction
When estimating a limit, we look at function values for inputs close to a target. At a domain endpoint, the inputs may be available only on one side of that target. The domain therefore tells us which direction of approach can be examined.
For example, if a function is defined on \([0,5]\), then values in its domain can approach \(0\) from the right, but not from the left. At \(5\), they can approach from the left, but not from the right. A graph or table may give strong evidence about the limit from the available side; it cannot provide evidence from a side where the function has no nearby inputs.
One-Sided Limits at an Endpoint
The notation \(x\to a^+\) means that \(x\) approaches \(a\) through values greater than \(a\). The notation \(x\to a^-\) means that \(x\) approaches \(a\) through values less than \(a\). In either case, the inputs must be in the function's domain.
For the usual interval domains in AP Calculus, the endpoint notation makes the available side clear. If the domain begins at \(a\), as in \([a,b]\), the approach at \(a\) is from the right. If the domain ends at \(b\), the approach at \(b\) is from the left.
A two-sided limit uses inputs approaching from both sides. At a left endpoint of \([a,b]\), there are no domain inputs just to the left of \(a\), so the available endpoint limit is the right-hand limit. At the right endpoint, the available endpoint limit is the left-hand limit. Do not invent values or behavior for the function outside its domain.
Estimating from a Graph
To read an endpoint limit from a graph, trace the plotted function only along the side of the endpoint that belongs to the domain. Follow the graph toward the endpoint and estimate the height it approaches. The function's value exactly at the endpoint, if shown, is a separate piece of information.
Worked Example: Read a Limit at the Left Endpoint
A function has domain \([0,4]\). Near \(x=0\), its graph passes through the points \((0.5,2.4)\), \((0.1,2.08)\), and \((0.01,2.008)\). A filled point at \(x=0\) has height \(7\). Estimate the endpoint limit.
Step 1: Find the available direction. Since the domain starts at \(0\), nearby domain inputs satisfy \(x>0\). The available approach is from the right, so the appropriate notation is \(\lim_{x\to0^+}f(x)\).
Step 2: Read the nearby heights. As \(x\) gets closer to \(0\) from above, the listed heights are \(2.4\), \(2.08\), and \(2.008\). Their distances from \(2\) are \(0.4\), \(0.08\), and \(0.008\):
The heights are getting closer to \(2\), so the graph suggests that the right-hand limit is \(2\).
Step 3: Separate the limit from the endpoint value. The filled point gives \(f(0)=7\), but it does not change the trend of nearby graph heights. The estimate is therefore \(\lim_{x\to0^+}f(x)=2\), while \(f(0)=7\).
Check: Each listed nearby height differs from \(2\) by the distances calculated above, and those distances shrink as the input approaches \(0\). The endpoint value \(7\) is not one of those nearby heights and is not the estimated limit.
Estimating from a Numerical Table
A table can show how function values behave for inputs approaching an endpoint. Use inputs from the domain, and check that they approach from the available side. For a left-hand approach, the inputs increase toward the target from below. For a right-hand approach, they decrease toward the target from above.
Worked Example: Use a Table at the Right Endpoint
A function has domain \((-\infty,3]\). A table gives values near its right endpoint:
| Input \(x\) | 2.9 | 2.99 | 2.999 |
|---|---|---|---|
| Value \(f(x)\) | 5.8 | 5.98 | 5.998 |
Estimate the endpoint limit.
Step 1: Determine the side. The domain contains inputs less than \(3\), but no inputs greater than \(3\). Thus the available approach is from the left, and we examine \(\lim_{x\to3^-}f(x)\).
Step 2: Compare the values with a candidate. The table's function values are close to \(6\). Their distances from \(6\) are
Meanwhile, the inputs' distances below \(3\) are
As the inputs get closer to \(3\) from below, the function values get closer to \(6\). The table suggests \(\lim_{x\to3^-}f(x)=6\).
Step 3: Check the estimate against each row. The three differences from the proposed limit are \(0.2\), \(0.02\), and \(0.002\), decreasing in order. This supports the estimate \(6\); the table does not give information about approaching from the right because that side is outside the domain.
Why Only the Available Side Counts
The one-sided limit is determined by function values at domain inputs near the target on the specified side. At an endpoint of an interval domain, all such nearby inputs lie on just one side. The other side is not merely unobserved in the graph or table: it is unavailable as part of the function's domain.
This explains why an endpoint problem should not be treated like an ordinary two-sided approach. For instance, at the left endpoint \(a\) of \([a,b]\), a table of inputs less than \(a\) cannot be used to estimate the limit of that function: those inputs do not belong to \([a,b]\). Instead, use inputs greater than \(a\) and closer to it.
When a Point Has No Available Approach
Being a boundary point of a domain does not automatically guarantee that domain inputs approach it. The inputs must get arbitrarily close. If a point is isolated in the domain, there are no nearby domain values to follow from either side, so a limit cannot be estimated from an approach within the domain.
Worked Example: Check Whether an Approach Exists
Suppose the domain is \(D=\{0\}\cup[2,4]\), and consider the endpoint \(x=0\). Is a one-sided limit at \(0\) available from the domain?
Step 1: Check the right side. The domain points near \(0\) include \(0\) itself, but the next domain inputs are at least \(2\). For example, there are no domain inputs in \((0,1)\). Thus there are not domain inputs greater than \(0\) arbitrarily close to \(0\).
Step 2: Check the left side. The domain contains no values less than \(0\), so there are no domain inputs approaching \(0\) from the left either.
Conclusion and check: Neither one-sided approach is available. The value \(f(0)\), if defined, would give information about the function at the single input \(0\), but no nearby domain values exist to establish a limit there.
A Reliable Endpoint Procedure
Use this sequence of checks whenever a graph or table concerns a domain endpoint:
Determine whether the domain begins or ends at the target, or whether the target is isolated.
Inputs greater than the left endpoint give a right-hand approach; inputs less than the right endpoint give a left-hand approach.
Use graph heights or table values from that side only, and look for a value they appear to approach.
Write the matching \(+\) or \(-\) notation and keep the limit distinct from the function value at the endpoint.
A Common Mistake
A frequent error is to report a two-sided limit at a left endpoint even though the domain provides inputs only from the right.
Wrong: If the domain is \([0,4]\) and the graph approaches height \(3\) as \(x\) approaches \(0\) along the curve, writing \(\lim_{x\to0}f(x)=3\) suggests an approach from both sides.
Correction: The graph supplies nearby inputs only for \(x>0\). Write \(\lim_{x\to0^+}f(x)=3\). There is no left-hand approach within this domain to combine with the right-hand evidence.
The notation is not a minor detail. It states which inputs support the estimate and prevents a conclusion from claiming more than the graph or table shows.
Check Your Understanding
For each question, identify the available side before estimating or interpreting the limit.
- A function has domain \([2,7]\). Which one-sided approach is available at \(x=2\), and which is available at \(x=7\)?
- A table lists inputs \(4.9\), \(4.99\), and \(4.999\) for a function whose domain ends at \(5\). Which one-sided limit can these inputs help estimate?
- A graph on domain \([0,3]\) approaches height \(1\) near \(x=0\), while \(f(0)=6\). What is the estimated one-sided limit, and what is the function value?
- Why can values from outside a function's domain not be used as evidence for an endpoint limit?
- If a domain contains \(0\) but no other points between \(0\) and \(2\), is a right-hand approach to \(0\) available within the domain? Explain.
