Reading a Graph Near an Input
A graph can show several kinds of behavior as \(x\) approaches a finite input \(a\). The curve might approach a fixed height, rise without bound, fall without bound, or fail to settle into any one behavior. To interpret a graph correctly, inspect the curve just to the left of \(a\) and just to the right of \(a\) separately.
The notation \(x\to a^-\) means that \(x\) approaches \(a\) through values less than \(a\). The notation \(x\to a^+\) means that \(x\) approaches \(a\) through values greater than \(a\). In both cases, the question concerns points near \(a\), not necessarily the value of the function at \(a\).
For a limit of \(-\infty\), the corresponding condition is that the function values eventually fall below every negative threshold: for every \(M>0\), \(f(x)<-M\) sufficiently near \(a\) on the indicated side. A finite one-sided limit \(L\) means that the nearby function values approach the fixed real number \(L\).
These definitions make the graph-reading language precise. A branch that rises without bound on the left indicates a left-hand limit of \(+\infty\); a branch that falls without bound on the right indicates a right-hand limit of \(-\infty\). A branch that settles toward height \(L\) indicates a finite limit \(L\). The function must have domain points arbitrarily close to \(a\) on the side being considered for that one-sided behavior to be read.
A Reliable Graph-Reading Process
For each side, follow the branch toward \(x=a\), not away from it. Ask what happens to the \(y\)-values as the horizontal distance to \(a\) shrinks. The height marked exactly at \(x=a\)—including a filled point, an open circle, or no point at all—does not answer this question.
Use the left branch for \(x\to a^-\) and the right branch for \(x\to a^+\).
Follow the graph in the direction of the input \(a\), observing the heights of points on that branch.
Record a finite height if the branch approaches one, \(+\infty\) if it rises without bound, \(-\infty\) if it falls without bound, or “does not exist” if it fails to settle into any one of these behaviors.
Use the two one-sided results to decide whether a two-sided limit exists.
A sketch is evidence of a trend, not a collection of exact measurements. A few plotted points with large \(y\)-values do not by themselves prove that a branch rises without bound. The graph should show a sustained trend toward larger values as the input approaches \(a\).
Worked Example: Opposite Infinite Behaviors
Worked Example: Opposite Infinite Behaviors
Suppose a graph has a vertical line at \(x=2\). As the left-hand branch approaches \(x=2\), it rises without bound. As the right-hand branch approaches \(x=2\), it falls without bound. A filled point is also shown at \((2,5)\). Find the one-sided limits and determine whether the two-sided limit exists.
Step 1: Read the left side. For \(x<2\), trace the branch toward \(x=2\). Its height increases without bound, so \(\lim_{x\to 2^-}f(x)=+\infty\).
Step 2: Read the right side. For \(x>2\), trace the branch toward \(x=2\). Its height decreases without bound, so \(\lim_{x\to 2^+}f(x)=-\infty\).
Step 3: Compare the results. The left side rises without bound, while the right side falls without bound. The two-sided limit therefore does not exist.
Check: The graph has \(f(2)=5\), but that value is not either one-sided limit and cannot make the two-sided limit exist. The nearby left branch rises and the nearby right branch falls, which confirms the classifications.
Worked Example: Matching Infinite Behavior
Worked Example: Matching Infinite Behavior
Suppose both branches near \(x=-1\) fall without bound as they approach \(x=-1\). The graph also has an open circle at \((-1,3)\). Determine the one-sided limits and describe the two-sided behavior.
Step 1: Read the left side. As \(x\to-1^-\), the branch falls without bound. Thus \(\lim_{x\to-1^-}f(x)=-\infty\).
Step 2: Read the right side. As \(x\to-1^+\), the branch also falls without bound. Thus \(\lim_{x\to-1^+}f(x)=-\infty\).
Step 3: Combine the sides. Both branches have the same unbounded behavior, so the graph indicates that \(f(x)\) falls without bound as \(x\to-1\). In extended-limit notation, \(\lim_{x\to-1}f(x)=-\infty\).
Check: Approaching \(-1\) from either direction gives the same downward trend. The open circle at height \(3\) does not affect that trend; it only marks a point not included on the graph.
Worked Example: One Finite Side and One Infinite Side
Worked Example: One Finite Side and One Infinite Side
Near \(x=4\), suppose the left branch approaches height \(2\), shown by an open circle at \((4,2)\). The right branch rises without bound as it approaches \(x=4\). Find both one-sided limits and determine whether a two-sided limit exists.
Step 1: Read the left side. The left branch settles toward the fixed height \(2\), so \(\lim_{x\to4^-}f(x)=2\).
Step 2: Read the right side. The right branch rises without bound, so \(\lim_{x\to4^+}f(x)=+\infty\).
Step 3: Compare the results. A finite value of \(2\) and \(+\infty\) are different behaviors. The two-sided limit does not exist.
Check: From the left, the graph gets close to height \(2\); from the right, its height exceeds every fixed bound sufficiently close to \(4\). Because these descriptions disagree, there is no single two-sided limiting behavior.
How One-Sided Limits Determine a Two-Sided Limit
The key rule is that a two-sided limit exists only when the two sides approach the same result. This applies to finite limits and to matching infinite behavior. The reason follows directly from what it means to approach \(a\) from both sides.
Justification. First suppose both one-sided limits equal the same finite number \(L\). For any \(\varepsilon>0\), the left-hand limit gives a distance \(\delta_1>0\) within which \(0<a-x<\delta_1\) ensures \(|f(x)-L|<\varepsilon\). The right-hand limit gives a distance \(\delta_2>0\) within which \(0<x-a<\delta_2\) ensures the same inequality. Set \(\delta=\min(\delta_1,\delta_2)\). Then \(0<|x-a|<\delta\) places \(x\) on one of these two sides, so \(|f(x)-L|<\varepsilon\). This is exactly the definition of the two-sided limit \(L\).
The same reasoning works for \(+\infty\): given any \(M>0\), the two one-sided limits provide distances \(\delta_1\) and \(\delta_2\) such that \(f(x)>M\) on their respective sides. With \(\delta=\min(\delta_1,\delta_2)\), every \(x\) satisfying \(0<|x-a|<\delta\) has \(f(x)>M\). Thus the two-sided behavior is \(+\infty\). Replacing \(f(x)>M\) by \(f(x)<-M\) proves the corresponding result for \(-\infty\).
Conversely, if a two-sided limit is \(L\), its defining condition applies to every \(x\) sufficiently close to \(a\) with \(x\ne a\), and therefore applies when \(x\) is restricted to either side. Both one-sided limits must equal \(L\). The same restriction argument applies to infinite behavior. Consequently, different one-sided results rule out a two-sided limit.
When a One-Sided Limit Does Not Exist
A branch need not approach a finite height or move steadily upward or downward. It may oscillate between heights, jump among separated values, or show another pattern that does not settle into one limiting behavior. In such a case, the one-sided limit does not exist.
Worked Example: Oscillation Near the Input
Suppose that as \(x\to 0^+\), a graph keeps oscillating between \(y=-1\) and \(y=1\), with neither height becoming the sole limiting value. On the left of \(0\), suppose the graph approaches height \(3\). Classify both sides and the two-sided limit.
Step 1: Read the right side. The branch continues to take values near both \(-1\) and \(1\) arbitrarily close to \(0\). It does not settle toward one finite number, nor does it rise or fall without bound. Therefore, \(\lim_{x\to0^+}f(x)\) does not exist.
Step 2: Read the left side. The left branch approaches \(3\), so \(\lim_{x\to0^-}f(x)=3\).
Step 3: Decide the two-sided result. Since the right-hand limit does not exist, there is no common two-sided limit.
Check: Arbitrarily close to \(0\) on the right, the graph still visits neighborhoods of both \(-1\) and \(1\). It therefore cannot remain close to one fixed number there. Nor can it exceed every positive bound or fall below every negative bound. The right-hand behavior is correctly classified as nonexistent.
Common Pitfall: Confusing the Point with the Approach
A frequent error is to read the height of the point at \(x=a\) instead of tracing the nearby graph. For example, suppose \(f(3)=7\), while the graph to the left of \(3\) rises without bound and the graph to the right falls without bound.
Incorrect conclusion: “The limit is \(7\), because \(f(3)=7\).” This uses a single point and ignores both approaching branches.
Correction: Read the two sides separately. The left-hand limit is \(+\infty\), and the right-hand limit is \(-\infty\). The two-sided limit does not exist. The value \(f(3)=7\) is compatible with those behaviors, but it does not determine them.
Another useful check is to make sure the direction of approach matches the branch being read. A branch to the left of \(a\) is used for \(x\to a^-\); a branch to the right is used for \(x\to a^+\). Do not reverse those labels just because the curve itself slopes upward or downward.
Why These Distinctions Matter
Graph-based limit questions test whether you can distinguish the behavior of a function near an input from its value at that input. They also require you to keep three questions separate: what happens from the left, what happens from the right, and whether those two behaviors agree. That separation is useful whenever a graph has a gap, a marked point, an unbounded branch, or irregular behavior near an input.
Check Your Understanding
For each question, focus on the branch behavior as the input approaches the specified value.
- A graph rises without bound on the left of \(x=1\) and rises without bound on the right. What are the two one-sided limits, and what is the two-sided behavior?
- As \(x\to 5^-\), a branch approaches height \(-2\); as \(x\to 5^+\), it falls without bound. Does the two-sided limit exist?
- A graph has \(f(0)=4\), while both nearby branches approach height \(1\). What is \(\lim_{x\to0}f(x)\), and why does \(f(0)\) not change it?
- On the right of \(x=2\), a graph continues to oscillate between two distinct heights arbitrarily close to \(2\). Is the right-hand limit finite, infinite, or nonexistent?
- State the condition on the one-sided limits that allows a finite two-sided limit to exist.
