What the Ends of a Graph Tell Us
A graph has two directions of end behavior: to the right, as \(x\) increases without bound, and to the left, as \(x\) decreases without bound. The function may approach a finite height in one direction, in both directions, or in neither. The two ends do not have to behave alike.
A horizontal asymptote describes one of these end behaviors. It is a horizontal line that the graph approaches as \(x\) tends to positive infinity or negative infinity. The line is identified by a finite limit; it is not found by evaluating the function at an “infinite” input.
Thus, one function can have two horizontal asymptotes, one for each end, and those lines can have different heights. It can also have the same horizontal asymptote at both ends, or have an asymptote at only one end. A finite limit is required for the corresponding end; behavior that grows without bound or continues to oscillate without settling to a single value does not give a horizontal asymptote there.
Reading a Limit as a Height
If \(\lim_{x\to\infty} f(x)=L\), the function values become arbitrarily close to \(L\) when \(x\) is sufficiently large and positive. The corresponding line is \(y=L\). If instead \(\lim_{x\to-\infty} f(x)=L\), then the values become arbitrarily close to \(L\) for sufficiently large negative magnitude of \(x\), and the line is again \(y=L\).
The direction matters. In \(\lim_{x\to\infty} f(x)\), we examine inputs far to the right. In \(\lim_{x\to-\infty} f(x)\), we examine inputs far to the left. A calculation for one limit alone does not establish what happens at the other end.
Why a Finite Limit Gives an Asymptote
The connection between limits and horizontal asymptotes follows directly from the meaning of a limit. For example, the statement \(\lim_{x\to\infty}f(x)=L\) means that for every positive tolerance \(\varepsilon\), there is a threshold \(M\) such that whenever \(x>M\), the function value differs from \(L\) by less than \(\varepsilon\). The difference \(|f(x)-L|\) is the vertical distance from the point on the graph to the line \(y=L\).
For the right end, suppose \(\lim_{x\to\infty} f(x)=L\), with \(L\) finite.
For every \(\varepsilon>0\), some \(M\) makes \(|f(x)-L|<\varepsilon\) whenever \(x>M\).
The inequality says that the graph lies within vertical distance \(\varepsilon\) of \(y=L\) for all sufficiently large \(x\). Since this holds for every positive tolerance, the graph approaches that horizontal line at the right end.
For the left end, the same reasoning uses the condition \(x<-M\) for a sufficiently large positive threshold \(M\). In that case, \(|f(x)-L|<\varepsilon\) means the graph gets arbitrarily close to \(y=L\) as \(x\) moves far to the left. This argument explains why the finite-limit definition and the geometric description of a horizontal asymptote agree.
Worked Example: The Same Asymptote at Both Ends
Worked Example: The Same Asymptote at Both Ends
Find the horizontal asymptote or asymptotes of \(f(x)=\frac{2x^2+3}{x^2-1}\) as \(x\to\infty\) and \(x\to-\infty\).
For either end, divide the numerator and denominator by \(x^2\), which is nonzero for sufficiently large \(|x|\):
As \(x\to\infty\), both \(\frac{3}{x^2}\) and \(\frac{1}{x^2}\) tend to \(0\). Therefore,
As \(x\to-\infty\), \(x^2\) also grows without bound, so the same two fractions tend to \(0\). Thus,
Both limits are finite and equal to \(2\). The horizontal asymptote at each end is \(y=2\). As a check, subtract the proposed height from the function:
Since \(\frac{5}{x^2-1}\to0\) at both ends, the function’s vertical distance from \(y=2\) tends to \(0\) in both directions, confirming the result.
Worked Example: Different Asymptotes at the Two Ends
Worked Example: Different Asymptotes at the Two Ends
Find the horizontal asymptote at each end for \(g(x)=\frac{x}{|x|+1}\).
The absolute value depends on the sign of \(x\), so treat the two directions separately. If \(x\to\infty\), then \(x>0\) eventually and \(|x|=x\). Hence,
As \(x\to\infty\), \(\frac{1}{x}\to0\), so
The right-end horizontal asymptote is \(y=1\).
For \(x\to-\infty\), we have \(x<0\) eventually and \(|x|=-x\). Therefore,
As \(x\to-\infty\), \(\frac{1}{x}\to0\), giving
The left-end horizontal asymptote is \(y=-1\). As a check, compare the function with each proposed height on its relevant end. For \(x>0\),
For \(x<0\),
The function approaches different finite heights at its two ends, so the two horizontal asymptotes are \(y=1\) and \(y=-1\).
Worked Example: An Asymptote at Only One End
Worked Example: An Asymptote at Only One End
Let \(h(x)=x^2\) for \(x\leq0\), and \(h(x)=4+\frac{1}{x}\) for \(x>0\). Determine whether there is a horizontal asymptote at each end.
When \(x\to\infty\), the applicable formula is \(h(x)=4+\frac{1}{x}\). Since \(\frac{1}{x}\to0\),
Thus \(y=4\) is a horizontal asymptote at the right end. To check the distance,
When \(x\to-\infty\), the applicable formula is \(h(x)=x^2\). Its values grow without bound, rather than approaching a finite number:
There is no horizontal asymptote at the left end. The conclusion is one-sided: the right end has asymptote \(y=4\), while the left end has no horizontal asymptote.
A Common Mistake: Treating Both Ends Alike
A frequent error is to simplify an expression using a sign assumption that is true at one end but false at the other. For instance, in the second example, someone might write \(|x|=x\) and then conclude that \(g(x)\to1\) as \(x\to-\infty\) as well. That substitution is incorrect: for negative \(x\), \(|x|=-x\), not \(x\).
The correction is to use the negative-\(x\) expression:
Before simplifying a formula involving \(|x|\), a piecewise definition, or another sign-sensitive expression, decide which inputs occur at the end you are analyzing. Then compute the positive- and negative-infinity limits separately.
A Reliable Procedure
When asked to find horizontal asymptotes from end behavior, organize the work around the two limits rather than guessing from the overall shape of the formula.
- Compute \(\lim_{x\to\infty}f(x)\). If it equals a finite number \(L\), state that \(y=L\) is a horizontal asymptote at the right end.
- Compute \(\lim_{x\to-\infty}f(x)\) separately. If it equals a finite number \(K\), state that \(y=K\) is a horizontal asymptote at the left end.
- If an end’s limit is infinite or does not exist, do not assign a horizontal asymptote at that end.
- Report both results, even if the limits agree. If they differ, name the corresponding end for each line.
For rational functions, the earlier material on limits at infinity for rational functions provides useful ways to compute the limits. Whatever method is used, the final asymptote claim still depends on a finite limit, and the positive and negative directions must be checked independently.
Check Your Understanding
For each question, keep the positive- and negative-infinity limits distinct.
- If \(\lim_{x\to\infty}f(x)=3\) and \(\lim_{x\to-\infty}f(x)=-2\), what horizontal asymptote occurs at each end?
- For \(p(x)=\frac{5x^2-1}{2x^2+4}\), find the horizontal asymptote at each end.
- For \(q(x)=\frac{x}{|x|+2}\), find the limits at positive and negative infinity and identify the two end asymptotes.
- If \(\lim_{x\to-\infty}r(x)=+\infty\), is there a horizontal asymptote at the left end? Explain using the definition.
- In one sentence, explain why finding the limit as \(x\to\infty\) does not by itself determine the left-end behavior.
